Limited Foresight Equilibrium
Jeevant Rampal∗†
February, 2017 (Work in Progress)
Keywords: Limited Foresight, Sequential Equilibrium, Centipede Game, Sequential Bar- gaining.
Abstract
This paper de nes the Limited Foresight Equilibrium (LFE). LFE provides an equi-
librium assesment for a model where players can possess limited foresight and they are
uncertain about the opponents' foresight while playing a nite dynamic game of per-
fect information. The LFE exists. LFE entails limited foresight players belief updating
about the opponents' foresight within the play of a game. LFE implies that the higher
the foresight of a player, the more accurate his beliefs about the opponents' foresights;
further, if a low foresight player nds himself at an unexpected position in the game,
he knows that one of his opponents has higher foresight than him. Thus, high foresight
types, in LFE, take reputation eects into account. In two applications, LFE is shown
to rationalize experimental ndings on the Bargaining game and the Centipede game.
∗The Ohio State University. Email: [email protected]. †I would like to thank James Peck for many important comments in both conceptualizing and writing this paper. I also want to acknowledge Yaron Azrieli, Paul Healy, Abhijit Banerji, and Anirban Kar for helpful comments and discussion. Last, I would like to thank the seminar participants at the Spring 2015 Midwest Economic Theory and International Trade Meetings, and the Winter School conference 2015 at DSE for useful and insightful comments.
1 1 Introduction
In a recent experimental study, Rampal (2017), tests a race game. One such race game used there is a two player game where players move alternately and remove 1, 2, or 3 items from a box containing 9 items at each move. The player who removes the last item loses. There is a second mover advantage in this game.1 The prize from winning as the rst-mover is 500 and the prize from winning as the second-mover is 200. Both players in a pair decide whether they want to be the rst/second mover. One of their choices is selected with 50 percent chance each. Note that the prize from winning as the rst mover is greater than the prize from winning as the rst mover, but winning as the rst mover is possible only if the second mover makes a mistake. Rampal (2017) found that if an expert player was told that his opponent was inexperienced, or if the expert player observed that his opponent made a mistake from a dominant position when playing against a computer (when it was strictly dominated to do so), the expert player was signi cantly more likely to choose to be the rst mover against such an opponent compared to an experienced opponent or an opponent who displayed perfect play. That is, Rampal (2017) nds that both (i) exogenous information, and (ii) endogenous inference about the opponent's inexperience increase the probability with which experienced players abandon the sure-win strategy (of being the second mover) and try for a higher payo attainable only by winning from a losing position, i.e., a position from which one wins only if the opponent makes a mistake. A maximum likelihood analysis shows that a model of limited foresight and uncertainty about the opponent's foresight as described in this paper explains the data better than the Dynamic Level-k (Ho and Su (2013)) and AQRE (McKelvey and Palfrey (1998)) models. The race game described above is a nite dynamic game of perfect information (not counting the simultaneous rst/second mover decision). Standard game theory uses the Subgame Perfect Nash Equilibrium, henceforth SPNE, to make predictions about outcomes in this class of games. But studies (cf. Rampal (2017), McKelvey and Palfrey (1992), Ochs
1The second mover should remove (4− opponent's previous choice) at each move to win.
2 and Roth (1989)) on the Centipede game (Rosenthal (1981)) and the Sequential Bargaining game (Rubinstein (1982) and Stahl (1973)), among others, have shown that SPNE can be a poor predictor of experimentally observed outcomes of games belonging to this class of nite, perfect information dynamic games. In any SPNE, each player of the game is supposed to use backward induction to calculate and choose his payo maximizing strategy, assuming that his opponents are also doing the same. These requirements generate two key reasons analyzed by the literature to try and explain SPNE's poor performance as a predictor of outcomes for games like the Centipede game and the Sequential Bargaining game. First, players may not always think backwards from their last move, i.e. they may have limited foresight and thus they may not perform perfect backward induction. This was shown by Johnson et al (2002) who monitored subjects using Mouselab during their study of three period bargaining; they found that the amount of periods subjects look forward was dierent across subjects.2 Second, it may not be common knowledge that all players can use perfect backward induction to calculate optimal strategies. Ignacio Palacios-Huerta and Oscar Volij (2009)3 and Levitt, List and Sado (2009) 4 explored the importance of this common knowledge about the ability of players to perform backward induction.5 In this paper, in de ning a new equilibrium concept, the Limited Foresight Equilibrium
2Johnson et al (2002) found that the amount of periods subjects look forward was a good predictor of subjects' actions. 3Ignacio Palacios-Huerta and Oscar Volij (2009) found that in 69% of the games, expert chess players in their sample stop a one-shot Centipede game at the rst node when matched with other chess players. This increases to 100% when looking at games where Grandmasters play against an opponent who is known to be an expert chess player. For the cases where an expert chess player knows that he/she is playing against a student subject, the frequency of stopping at the rst node drops. They attribute this result to perfect common knowledge of rationality among capable players, and thus conclude that it is the level of rationality and information about opponent's rationality that determines outcomes rather than altruism or social preference. 4Levitt, List and Sado (2011) test the performance of chess players with Elo rating between 1789 and 2367 in the Centipede game and two zero sum games requiring 10 steps of backward induction. They nd that the chess players play like student subjects in the Centipede game, cooperating in the beginning. As we show later, even an arbitrarily small probability on limited foresight can cause this result. 5For an epistemic discussion of backward induction see Aumann (1995), Battigalli (1997), Ben-Porath (1997), Binmore (1996), and Brandenburger and Friedenberg (2014). Bonanno (2001) studies backward induction in terms of temporal logic.
3 (LFE), we theoretically model two features: (i) limited foresight, i.e. a limited ability to do backward induction, and (ii) uncertainty about the opponents' foresight. Foresight level as de ned in this paper implies the number of subsequent staged that a player can observe from his move in the game. If a player's foresight level is not enough to perceive the entire game tree, then a player is said to have limited foresight. One key question in modeling limited foresight is that if a limited foresight player cannot observe until the last stage of a dynamic game, what payo pro les does he use to make choices that are optimal given his limited foresight? Here we use the ad-hoc rule used by Mantovani (2014), which speci es that the payo pro les that the limited foresight player observes at the end of his foresight are a function of the actual payos possible in the game from the stage at which his foresight ends. Ke (2017) explores which function satis es certain desirable axioms. The second feature of our model is uncertainty and updating about one's opponents' levels of foresight within the play of a nite dynamic game of seemingly perfect information. For example, the race game tested in Rampal (2017) was setup as a game of perfect information. Yet, we found that expert players were systematically deviating from the SPNE strategy based on their belief about the opponent's expertise.6 Thus, to model uncertainty about each player's foresight-level, we map the underlying game of seemingly perfect information to a game of imperfect information called an Interaction game. In the interaction game, each player can have one of several possible types, which is private information. Each type denotes a particular level of foresight, ranging from 0 foresight to full foresight. The only imperfection in the information of each player-type is they are not sure about the question what is the type/foresight-level of my opponents? In particular, all the previous moves of all the opponents are known. Nature's probability distribution on all possible player-type combinations is assumed to be common knowledge. The Interaction game nests the underlying game of perfect information as the case where the probability of all players being only the full foresight type is 1. The LFE provides a belief and strategy pro le for the
6These deviations also proved to be ex-post optimal.
4 Interaction game. That is, the LFE provides an action prediction and a belief prediction at every move of each player-type. The Interaction game does not model limited foresight per-se. That is, we cannot solve for the optimal action of a limited foresight player-type using the Interaction game because a limited foresight player-type can only observe a curtailed version of the Interaction game. Thus, we consider appropriately curtailed versions of the Interaction game, called Curtailed games, which are observed by the limited foresight player-types at their moves to solve for their optimal actions and beliefs. If an S staged Interaction Game is curtailed after the nth stage actions to generate Curtailed Game(n) or CG(n), where n = 1, 2, ..., S. The LFE is solved and de ned recursively. We start with the 1-staged Curtailed Game, named CG(1), and solve for its Sequential Equilibrium (SE), Kreps and Wilson (1982), to obtain the rst player's 0 foresight type's rst stage LFE action and beliefs. Next, taking this action and belief as given, we solve for the SE of CG(2) solving for the LFE actions and beliefs of rst player's 1 foresight type's information set at stage 1, and the second player's 0 foresight type's information set at stage 2; as they both observe CG(2) at those information sets. We proceed stepwise to obtain the LFE actions and beliefs for all the information sets of the Interaction Game. De ning LFE as above provides us with several desirable properties. First, the LFE exists and it is upperhemicontinous. Second, higher foresight types correctly anticipate lower types' moves. Reynolds (1992) testing recognition of opponent's expertise among chess players found that Higher rated players consistently made lower estimation errors (of chess players' ELO ratings). Rampal (2017) and Reynolds (1992) both found that the estimation error (about the opponent's expertise) decreased as a function of the number of moves revealed, a second property that holds in our model when high foresight types are estimating which lower opponents' types they are playing against. The third property of LFE is that, if a low foresight type observes actions that were not part of the LFE strategies of lower foresight opponent types, then he discovers that he is playing against some higher total
5 foresight type. This feature follows from the recursive de nition of LFE and a restriction on the beliefs. Last, any foresight type does the best he can within the bound of his foresight, given his belief about the probability distribution on opponents' types. The attempt by lower types to recognize opponent type and adjust behavior implies the fourth feature of our model: reputation eects. High types have to decide on what's optimal: pretending to be a low type or revealing their type. Importantly, all this learning and updating happens within a play of the game. In summary, LFE requires that at each move, limited foresight types behave in a sequentially rational manner, given the belief about opponents' types and their own limited understanding of the game. These features imply passing until the last few stages in a Centipede game with more than 4 stages, for arbitrary probability on limited foresight. LFE is also shown to have the ability to explain delays, near equal splits, disadvantageous counter proposals and subgame inconsistency, all of which have been observed in the experimental studies on Sequential Bargaining. The Valuation Equilibrium by Jehiel and Samet (2007) models how cognitively con- strained players may group nodes of a sequential move game into exogenously given similarity classes, where each similarity class has a given valuation. Though related, Valuation Equi- librium does not deal with limited foresight speci cally. Jehiel (1995) de nes the Limited Forecast Equilibrium (LFE), where each player, at each of his moves, chooses his strategy to maximize his average payo within his foresight horizon, given his forecast about the upcom- ing moves within that horizon. The forecasts are constrained to be consistent with the (LFE) strategies. Jehiel (1998a) provides a learning justi cation for these forecasts. Jehiel (1998b) and Jehiel (2001) extend the LFE to repeated games. The closest works to ours in the area of dynamic games are the working papers of Mantovani (2014), Ke (2017), (discussed above) and Roomets (2010). These papers, in independent projects, model limited foresight in a similar fashion to ours. Additionally, Mantovani (2014) endogenizes the choice of foresight. He also demonstrates the existence of limited foresight using an experiment on a dierent
6 race game. However, the second feature of our model, the uncertainty about the oppo- nents' foresight type, is absent from all the papers mentioned above. Consequently, learning about opponents' types within the game, strategic adjustments after updating beliefs about opponents' foresight and reputation eects do not feature in these papers. Most of the related Level-k literature deals with simultaneous move games (Stahl and Wilson (1995); Nagel (1995); Stahl (1996); Ho et al. (1998); Costa-Gomes et al. (2001); Camerer et al. (2004); Costa-Gomes and Crawford (2006); Crawford and Iriberri (2007a, b)), but it does capture the uncertainty about the opponent's expertise. Aloui and Penta (2016) endogenize the choice of level in a Level-k framework by modeling and studying the incentives and costs of choosing a certain cognitive level. In a simultaneous move setting, these incentives and costs are shown to depend on one's own payo, the opponent's level, and even how the opponent's level is aected by the opponent's payo. They disentangle the eect of one's own cognitive limitation from the eect of one's beliefs about the opponent's cognition. Experimental studies of the eect of the opponent's cognitive level over a player's choices include Agranov et al (2012), Gill and Prowse (2014), and Slonim (2005), among others. Ho and Su (2012) adapted the Level-k model for sequential move games. They allow for learning across repetitions of play of the same game as opposed to within the play of a game as in here. They apply their model to t data from experiments on the Centipede game and the Sequential Bargaining game. Kawagoe and Takizawa (2012) also study the dynamic Centipede Game using level-k using a logit error structure. The key dierence in Level-k models and models of limited foresight like ours is that the Level-k models assume that players are rational but possess subjective beliefs about the rationality of the opponent, while models of limited foresight allow players to be bounded rational because of limitations in foresight. In particular, a model of limited foresight can allow players to play a dominated strategy in a dynamic game due to limited foresight (eg. in Rampal (2017), in one of the treatments, about 40 percent of the subjects make a mistake in the race game as the
7 second mover when doing so is dominated), however, the Level-k model will not allow any level above Level-0 to choose the dominated strategy. The AQRE model of McKelvey and Palfrey (1998) uses a model where players are playing error prone strategies. This approach is eective in data tting but does not model limited foresight, which produces peculiar patterns of mistakes7, nor does it model uncertainty and updating about the opponent's foresight. The reputation eects in our model are similar to the crazy type literature started by Kreps, Wilson, Milgrom and Roberts in 1982, yet there are important dierences. Their crazy types' behavior is exogenous, and their crazy types, whose counterparts in the LFE model would be the player-types with low foresight, have no incentive to discover whom they are playing against.
2 Model
The model that we de ne here seeks to capture the scenario where a nite set of players are playing what seems to be a nite dynamic game of perfect information. Speci cally, all prior actions taken in the game are observed by every player at his move, but every player has a particular level of expertise/expereince in the game and every player is uncertain about each opponent's level of expertise/experience in the game. In particular, we will focus on the case where this level of expertise/experience translates into a level of foresight. That is we will model the scenario where every player can have one of various possible levels of foresight and every player is uncertain about the level of foresight of each of his opponents.8 The foresight level of a player is de ned as the number of subsequent stages that a player can observe from any given move. To model this scenario we start with the game that the players seem to be playing, i.e. a nite dynamic game of perfect information called Γ0. It is helpful to think of Γ0 as the 7For example in Rampal (2017) we observed a high proportion of dominated choices with a lot of stages left and a much lower proportion of dominated choices with fewer stages left. 8We don't model how the expertise/experience translates into a particular level of foresight. This is left for future research.
8 game of perfect information that the experimenter sets up for a set of players to play. We map this game to the game that is actually being played, i.e., a standard Bayesian game of imperfect information called Γ where every player in Γ0 can have one of several possible types where each type denotes a particular level of foresight. For example, in the experiment in Rampal (2017), players with dierent levels of experience in a race game were playing in pairs. Subjects were aware that there was variation in the experience-level across subjects. Rampal (2017) found that both endogenous inference and exogenous information about the opponent's experience-level aected the observed optimal choices.9
In Γ, each player-type is uncertain about each opponent's level of foresight or, equiva- lently, each opponent's type. The Limited Foresight Equilibrium (henceforth LFE) provides a strategy and a belief pro le for Γ. However, we cannot solve for the LFE actions and beliefs of player-types who have limited foresight using Γ because player-types with limited foresight cannot observe Γ at their moves. Therefore we will consider appropriately curtailed versions of Γ, which is what the limited foresight player-types observe, to solve for the LFE strategy and belief pro le of Γ. In subsection 2.1 we de ne the underlying perfect information game
Γ0. In 2.2 we construct Γ, the game of imperfect information, from Γ0. In 2.3 we construct the curtailed versions of Γ which are observed by player-types possessing limited foresight.
2.1 The Underlying Perfect Information Game
We use the standard notation from Osborne and Rubinstein (1994), with minor modi ca-
10 tions, to de ne an extensive game with perfect information and perfect recall called Γ0. In particular, Γ0 is de ned as a collection of the following components. 9One of the games used there is a two player game where players move alternately and remove 1, 2, or 3 items from a box containing 9 items at each move. The player who removes the last item loses. There is a second mover advantage in this game. The prize from winning as the rst-mover is 500 and the prize from winning as the second-mover is 200. Both players in a pair decide whether they want to be the rst/second mover. One of their choices is selected with 50 percent chance each. Rampal (2017) found that if an expert player was told that his opponent was inexperienced, or if the expert player observed that his opponent messed up the play from a dominant position when playing against a computer, the expert player was signi cantly more likely to choose to be the rst mover against such an opponent compared to an
9 1
P 1 T 1 2 4, 1
P 2 T 2 1 2, 8
P 3 T 3
8, 32 16, 4
Figure 1: An Underlying Game of Perfect Information-Three Staged Centipede Game
• A set of players N0.
• A set H0 of nite sequences or histories such that:
The empty sequence ∅ is a member of H0. We refer to ∅ as the initial history.
If k and then k . Where the th action, k, (a0)k=1,...,K ∈ H0 L < K (a0)k=1,...,L ∈ H0 k a0 is said to be taken at the kth stage of the game. The set of terminal histories,
denoted as , is de ned as the set of histories k such that there Z0 (a0)k=1,...,K ∈ H0 is no such that k . (K + 1) (a0)k=1,...,K+1 ∈ H0
• A set of possible actions in the game, A0, and an action correspondence A0(.) which
maps h0 ∈ H0 to a set A0(h0) ≡ {a0 :(h0, a0) ∈ H0}.
• A function P0(.), called the player function, which maps each element of H0 to an
element in N0. That is, P0 assigns a player P0(h0) to each history h0.
• For each player i ∈ N0, a Bernouli utility function ui which maps terminal histories to
real numbers, i.e., ui maps a terminal history z0 ∈ Z0 to a payo ui(z0) ∈ R. experienced opponent or an opponent who displayed perfect play. 10We don't specify the conditions for perfect recall.
10 Thus, is de ned as . Let the maximum number of stages in be Γ0 {N0,H0,P0,A0, {ui}i∈N0 } Γ0 S.11 For example. consider the three-staged centipede game as an example of an underlying game of perfect information.
2.2 Constructing a Game of Imperfect Information from the Un-
derlying Game of Perfect Information
We now construct an extensive game of imperfect information called Γ from the extensive game of perfect information, Γ0. The only form of imperfection in information we allow in Γ relative to Γ0 comes from the feature that we seek to model a scenario where each player i ∈ N0 has several possible types and each player's type is his private information. In particular except Nature's move determining the probability distribution on players' types, all other prior actions will be known at each move. For example, consider the case where
N0 = {Ann, Bob} are playing Γ0 in the underlying game. To de ne Γ, in the very rst stage of Γ, we will introduce Nature's move which speci es the probability distribution over the possible combinations of types of the players in N0. Roughly speaking, after Nature moves, Ann and Bob will play Γ0 knowing his/her own type, but without knowing his/her opponent's type. For example, suppose Γ0 is tic-tac-toe. Suppose that Nature speci es that
0 type tAnn of Ann has a 30 percent chance of occurring and type tAnn of Ann has a 70 percent chance of occurring. Independently, type tBob of Bob has a 60 percent chance of occurring
0 and type tBob of Bob has a 40 percent chance of occurring. Then, in Γ, after this Nature's
0 move, Ann will play tic-tac-toe with Bob knowing her type, tAnn or tAnn, whichever it may
0 be, but without knowing if Bob's type is tBob or tBob. We know proceed to formally de ning the construction of Γ from Γ0. The extensive game of imperfect information Γ, constructed from Γ, has the following components.
• A set of player-types termed N. To construct N from N0, each i ∈ N0 generates a
corresponding set of i-types in Γ called Ti. Let si denote the rst stage of Γ0 at which 11Formally, : k S ≡ max{K (a0 )k=1,...,K ∈ H0}.
11 12 i moves, then the set of possible types of player i in Γ is given by Ti ≡ {0i, 1i, .., (S − . Further, a particular combination of player-types of is . Then si)i} Γ t ∈ T = ×i∈N0 Ti N = S T . A player-type, for example 1 , should be interpreted as follows. 1 is i∈No i i i
type 1 of player i ∈ N0. 1i is that type of player i who has foresight level of 1, i.e.,
given his move in Γ, 1i can observe one more stage after his move. We model limited foresight more comprehensively in subsection 2.3.
• A set of sequences H, and a set of terminal sequences Z. The sets H and Z are
generated from H0 using a set valued mapping Seq: H0 =⇒ H. Each sequence (ak) ∈ H corresponds to a set of sequences in H, and H = S Seq(h ). 0 k=1,...,K 0 h0∈H0 0 The elements of k comprise of all possible combinations of types of Seq((a0)k=1,...,K ) each of the players of chosen by Nature at the history before k . That Γ0 ∅ (a0)k=1,...,K is, k k : .13 Seq((a0)k=1,...,K ) = {(t, (a0)k=1,...,K ) t ∈ T }
• A player function P , which maps each element of H\Z to an element in N. The function P has the following properties.
P (∅) =Nature.
If player moved after k then some moves after k i (a0)k=1,...,K ∈ H0 ti ∈ Ti ((ti, t−i), (a0)k=1,...,K ) ∈ . That is, consider an arbitrary k . If k H ((ti, t−i), (a0)k=1,...,K ) ∈ H P0((a0)k=1,...,K ) = , then k . i P (((ti, t−i), (a0)k=1,...,K )) = ti
• Nature's move which speci es a probability distribution on T . This distribution, de- noted by ρ, is assumed to be common knowledge.14 ρ(t) ∈ [0, 1] for all t ∈ T , and P . t∈T ρ(t) = 1
• A set of possible actions in the game, A, and an action correspondence A(.).
12Formally, : k and k si ≡ min{K (a0 )k=1,...,K−1 ∈ H0 P0((a0 )k=1,...,K−1) = i} 13We say that the Nature's ∅ history action is taken at the 0th stage. 14The common knowledge assumption helps simplify a lot of the following analysis. We discuss the eect of weakening of this assumption later.
12 A(.) maps h ∈ H to a set A(h) ≡ {a :(h, a) ∈ H}.
The set of possible actions, or action set, after a sequence h ∈ H is the same
as the action set after the corresponding h0 ∈ H0 that generated h. That is, consider an arbitrary such that . Let k . Then h ∈ H h 6= ∅ h = (t, (a0)k=1,...,K )) k k . A((t, (a0)k=1,...,K )) = A0((a0)k=1,...,K )
• For each player type ti, a partition I(ti) of {h ∈ H: P (h) = ti}. I(ti) ∈ I(ti) is
an information set of ti. These information sets obey the usual restriction that the actions available from and the player moving at all histories of an information set must
15 be the same. The construction of Γ from Γ0 gives us more structure. Consider an
arbitrary history h0 of Γ0. Suppose i is the player moving after h0, that is, P0(h0) = i.
Then h0 will map to Seq(h0) = {(t, h0): t ∈ T } in Γ. The set {(t, h0): t ∈ T } will
be subdivided into |Ti| information sets in Γ, one information set for each ti ∈ Ti.
−1 Further, for each ti, his information set I(ti) such that Seq (I(ti)) = h0 is given by
{((ti, t−i), h0): t−i ∈ T−i}. That is, at each such information set of Γ, the player-type
moving there, ti, is aware about all prior actions, but he is uncertain about which combination of opponents' types, t , he is playing against. As S Seq(h ) = H, −i h0∈H0 0 all the information sets of Γ obey this structure.
For each player-type , a Bernouli utility function which maps terminal • ti ∈ N uti histories Z to real numbers. Additionally, for each z ∈ Z, the utility derived by an arbitrary player-type at , denoted as , is equal to the utility derived by at ti z uti (z) i the corresponding . That is, −1 , , , and for z0 ∈ Z0 uti (z) = ui(Seq (z)) ∀ ti ∈ Ti ∀ i ∈ N0 all z ∈ Z.
Thus, , corresponding to , is de ned by its con- Γ = {N,H, {I(ti)}ti∈N , P, A, {uti }ti∈N } Γ0 struction using the Seq(.) correspondence. As an example, consider the conversion of the Centipede game in gure 1 to an Interaction game.
15 0 0 Formally, P (h) = ti ∀h ∈ I(ii), and A(h) = A(h ) ∀h, h ∈ I(ti).
13 Nature
(01, 02) (01, 12) (11, 02) (11, 12) (21, 02) (21, 12) 01 01 11 11 21 21
P 1 T 1 P 1 T 1 P 1 T 1 P 1 T 1 P 1 T 1 P 1 T 1
02 12 02 12 02 12 4, 1 4, 1 4, 1 4, 1 4, 1 4, 1
P 2 T 2 P 2 T 2 P 2 T 2 P 2 T 2 P 2 T 2 P 2 T 2
01 01 11 11 21 21 2, 8 2, 8 2, 8 2, 8 2, 8 2, 8
P 3 T 3 P 3 T 3 P 3 T 3 P 3 T 3 P 3 T 3 P 3 T 3
8, 32 16, 4 8, 32 16, 4 8, 32 16, 4 8, 32 16, 4 8, 32 16, 4 8, 32 16, 4
Figure 2: The Centipede Game Converted to an Interaction Game
Notes. The gure shows the conversion of the Γ0 (the Centipede game depicted in Figure 1) into an Interaction game, Γ, depicted here. As there are 3 stages in the Centipede game in Figure 1,
There are 3 types of player 1 possible, {01, 11, 21}, and two types of player 2 possible, {02, 12}.
For each combination of player 1's type and player 2's type (who could be playing Γ0 with each
other) we redraw Γ0 to generate Γ. We construct the information sets so that each player 1 type, at each of his moves, observes the sequence of prior actions played, but he doesn't know which
player 2 type (02 or 12) he is playing against. Similarly, each player 2 type, at each of his moves,
observes the sequence of prior actions played, but he doesn't know which player 1 type (01, 11 or
21) he is playing against. 2.3 What Limited Foresight Types Observe: Curtailed Games
The Limited Foresight Equilibrium (LFE) provides an outcome prediction for the Interaction game Γ by specifying an equilibrium strategy pro le and the associated belief pro le for it. We put equilibrium in quotes because LFE cannot be solved using the Interaction game. We can't use a solution concept directly on the Interaction game because limited foresight player-types cannot observe the Interaction game at all of their information sets, hence
14 cannot be optimizing based on it. Limited-foresight player-types, at each move, optimize based on a curtailed version of the Interaction Game that they observe from that move given their limited foresight level. That is, player-types use their move speci c curtailed version of the Interaction game to optimize. These curtailed versions of the Interaction game are said to be the Curtailed games generated from an Interaction game. As the name suggests, a Curtailed game is de ned by curtailing the Interaction game at a particular stage. Consider an Interaction Game with S-stages, Γ. An n staged Curtailed Game constructed from Γ will be labeled as CG(n). Let
CG(n) = {N,Hn, {In(t )} ,P n,An, {un } }. i ti∈N ti ti∈N
The components of CG(n) are de ned as follows.
• CG(n) is an exact replica of Γ until (and including) stage (n − 1). The player set N of CG(n) is the same as the player set N of Γ. Hn, the set of histories of Γn, is de ned
as k such that . Further, n is partitioned into information (t, (a0)k=1,...,K ) ∈ H K ≤ n H sets, n , exactly as is partitioned into . The player function {I (ti)}ti∈N H {I(ti)}ti∈N P n(h) = P (h), and the action function An(h) = A(h), for all h ∈ Hn T H. Nature's common-knowledge prior distribution over T , given by ρ in Γ, is the same in every CG(n) and it remains common knowledge in every CG(n).16
• The set of terminal histories of CG(n), denoted as Zn, contains two kinds of terminal histories. First, the terminal histories of Γ which end at or before an nth stage action;
formally, let n k : }. Second, those sequences/histo- Z (1) = {(t, (a0)k=1,...,K ) ∈ Z K ≤ n ries k with , which are curtailed at k and are (t, (a0)k=1,...,K ) ∈ H K > n (t, (a0)k=1,...,n) converted to terminal histories, n , in . Formally, n k Z (2) CG(n) Z (2) = {(t, (a0)k=1,...,n) ∈ 16The common knowledge assumption helps in reducing the number of possible dierent ways the various limited foresight players can observe a curtailed version of the Interaction Game. If every player-type had a dierent subjective belief over opponents' types, t−i, then we would have to contruct a move speci c Curtailed Game for each individual player-type. The common knowledge prior distribution lets us consider only S possible curtailed versions of Γ for the purpose of solving for the strategies and beliefs of all the player-types. This will become clearer when we de ne LFE.
15 H\Z}. The set of terminal histories of CG(n) is denoted as Zn = Zn(1) S Zn(2).
• For those terminal histories of CG(n) which are also the terminal histories of Γ, the payos of each player-type remain the same. That is, for all zn ∈ Z T Zn, un (zn) = ti n . The controversial choice that must be made in curtailing the Interaction uti (z ) Game is that what is the payo pro le associated with a terminal history of CG(n) which is not a terminal history of Γ? Any payo numbers placed at such synthetic terminal histories, Zn(2), of CG(n), will have to follow some ad-hoc rule.17We use the [(min + max) ÷ 2] rule of Mantovani (2014).18 The [(min + max) ÷ 2] rule implies that each player-type's payo after h in CG(n) is the average of the minimum and the maximum that that player-type could achieve in Γ following all possible terminal action sequences after h. That is, for each h ∈ Zn(2), let Z(h) be the set of terminal histories of Γ where the actions in the rst n stages are played as speci ed in h. Formally, let Z(h) = {z ∈ Z: z = (h, (ak) )}. Then, for each t ∈ N, un (.) is de ned over 0 k=n+1,...,K i ti Zn(2) as follows.
min{u (z): z ∈ Z(h)} + max{u (z): z ∈ Z(h)} un (h) = ti ti for all h ∈ Zn(2) (1) ti 2
As an example of a curtailed game, consider the one-staged curtailed game, CG(1) depcited in Figure 3, constructed from the Interaction game in Figure 2.
17Shaowei Ke (2017) has a working paper that justi es his rule using axiomatic foundations. To be clear, our contribution is to model limited foresight with uncertainty and updating about the opponent's foresight within a play of the game. 18In an older version of our paper we used a mean of stagewise means rule explained there, which didn't change any of the results that follow.
16 Nature
(01, 02) (01, 12) (11, 02) (11, 12) (21, 02) (21, 12) 1 01(D ) 11 21
P 1∗ T 1 P 1∗ T 1 P 1 T 1 P 1 T 1 P 1 T 1 P 1 T 1
9, 18 4, 1 9, 18 4, 1 9, 18 4, 1 9, 18 4, 1 9, 18 4, 1 9, 18 4, 1
Figure 3: Curtailed Game (1)
Notes. The gure shows the conversion of Γ, the Interaction game depicted in Figure 2, into its shortest curtailed game, CG(1), depicted here. CG(1) is identical to Γ in all respects except that CG(1) ends after the rst stage action. If any type of player 1 chooses T 1 (take in stage 1) in the rst stage then the associated CG(1) payo pro le is (4, 1), as in the Interaction game. However, if some player 1 type chooses P 1 (pass in stage 1), then in the construction of CG(1), we must curtail the Interaction game after and use the min+max rule for payos. For example, after P 1 2 playing P 1, the maximum a player 1 type can get in Γ is 16 and the minimum he can get is 2, thus, his payo from choosing in is min+max . We mark 0 rst stage information P 1 CG(1) 2 = 9 01s 1 set as D . This is because CG(1) is exactly what 01 observes at his rst stage information set.
Thus, CG(1) is decisive for 01 at stage 1. This will be made precise when we de ne LFE. 3 Limited Foresight Equilibrium
We now proceed to de ning the LFE. First we need to de ne total foresight. Let a limited foresight type's total foresight be the sum of (i) the stage number that the limited foresight type is moving at, and (ii) the level of foresight of that limited foresight type. Suppose we are trying to solve for the LFE action of the limited foresight player-type ti moving at some information set I(ti). The de nition of LFE boils down to three rules of thumb: (a) ti knows the LFE actions of all the player types with lesser total foresight than him; (b) ti assumes that equal or higher total foresight types, including ti himself, together choose a strategy pro le for the curtailed game that he observes at I(ti). This strategy pro le must be sequentially rational for each player type in this curtailed game given (a), the rest of the strategy pro le, and the beliefs of these player types. (c) 0 beliefs, and the beliefs of tis
17 all other player-types in this curtailed game are calculated using the Bayes' rule, given the strategy pro le in (b) and (a). Formally, we solve for ti's LFE action and belief at I(ti) be solving for the Sequential Equilibrium of the curtailed game he observes at I(ti), after taking the LFE actions in (a) given as Nature's moves.
We know that ti at I(ti) observed a particular curtailed version of the Interaction game.
To save us eort, we look for all other player-types, say tj, tk and their information sets
I(tj) and I(tk), such that tj, tk also observed exactly the same curtailed game as ti at I(ti).
We note the actions of ti, tj, and tk at I(ti), I(tj), and I(tk) solved for in (b) and their beliefs solved for in (c) as the LFE actions and beliefs at these information sets. Proceeding from the shortest curtailed game to the Interaction game as above gives us the LFE for an Interaction game. Note that in solving for the Sequential Equilibrium of a curtailed game, we are also solving the LFE actions and beliefs of player-types other than ti, tj, and tk at
I(ti), I(tj), and I(tk). However, we do not count them as LFE actions or beliefs. They are simply needed to calculate the LFE actions and beliefs of ti, tj, and tk at I(ti), I(tj), and
I(tk).
3.1 De ning the Limited Foresight Equilibrium
To de ne the LFE, we will rst need two de nitions: total foresight and decisive information sets. To de ne total foresight, let the foresight level of player-type ti be denoted as ti itself.
For example, the player-type 3i has a foresight level of 3. We denote the foresight level of 3i
19 as 3i, it is understood that the foresight level is actually 3. De nition 1 (Total Foresight): Consider a sequence k , and h = (t, (a0)k=1,...,s−1) ∈ H
th the player-type P (h) moving at the s stage of Γ. Let P (h) = ti. The total foresight of player-type ti at stage s is (ti + s). De nition 2 (Decisive Information Sets): Let Γ be an S-staged Interaction Game. An n-staged Curtailed Game, CG(n), where n < S is said to be decisive for the information sets
19 So 3i + 4 = 7. Believe us, this abuse of notation helps simplify the notation.
18 n n D of CG(n), i for all I(ti) ∈ D , if I(ti) occurs at stage s of CG(n), then we must have that the total foresight of ti at stage s is equal to n, that is, (ti + s = n) should hold true. is decisive for S SS−1 n. We also say that the information CG(S) = Γ D = {I(ti)}ti∈N − n=1 D sets in Dn are decisive for CG(n). Consider an S-staged Interaction Game Γ. Γ generates S distinct Curtailed Games given the assumption that Nature's distribution over T is common knowledge in each Curtailed Game. Let M be the number of player-types in Γ. Denote a strategy pro le of Γ as π. Where . Denote a belief system of as . Where . For each π = ((πti )ti∈N ) Γ µ µ = ((µti )ti∈N ) player-type , speci es the action choice and belief of at all the information sets ti (πti , µti ) ti of Γ where ti moves. Formally, consider an arbitrary information set I(ti) of ti. Let I(ti) be generated by . That is, −1 . Then and h0 ∈ H0 Seq (I(ti)) = h0 πti : I(ti) 7−→ ∆(A(I(ti))), : . µti : I(ti) 7−→ ∆{((ti, t−i), h0) t−i ∈ T−i} Consider an n ∈ {1, ...S}. Let (σn, bn) denote an assessment for CG(n). That is, σn = ((σn ) ) and bn = ((bn ) ) denote a strategy and belief pro le for CG(n) respectively. ti ti∈N ti ti∈N For each player-type t , (σn , bn ) speci es the action choice and belief of player-type t at all i ti ti i the information sets of CG(n) where ti moves. Let the set of sequential equilibria of any game G be denoted as Ψ(G). Let Dn denote the decisive information sets of CG(n). The Limited Foresight Equilibrium of Γ will be an assesment (π, µ) for Γ that we will construct below. We need one more de nition before de ning an LFE.
De nition 3 (Modi ed Curtailed Games): Consider a curtailed game CG(n) for some n ∈ {2, .., S}. Suppose π(.) provides the LFE strategy pro le for all the decisive information sets of through , Sn−1 k. Then is de ned by its construction CG(1) CG(n − 1) k=1 D MCG(n) from CG(n), given π, by making two modi cations. First, modify the player function of CG(n), P n, to mP n so that in MCG(n), for all the decisive information sets of CG(1) through CG(n−1), the player-type moving there is replaced by Nature. That is, in MCG(n),
n for all the information sets Sn−1 k. For all Sn−1 k, n mP (I) = Nature I ∈ k=1 D I/∈ k=1 D mP (I) = P n(I). Second, we specify how Nature moves at these information sets using ρn, which is an
19 augmented version of the Nature's move inCG(n), given by ρ. In particular, in MCG(n), the initial prior distribution is the same as CG(n) and the Interaction Game Γ, that is, ρn(∅) = ρ.
Further, for all Sn−1 k, n , that is, for all the decisive information sets of I ∈ k=1 D ρ (I) = π(I) CG(1) through CG(n − 1), Nature moves exactly as speci ed by π. Given the notation and de nitions above, we have the following de nition of LFE:
De nition 4 : (π, µ) is a Limited Foresight Equilibrium of an S-staged Interaction Game Γ if it is constructed in the following S steps: Step 1: Select a Sequential Equilibrium assessment (σ1, b1) ∈ Ψ(CG(1)). Set (π(I), µ(I)) = (σ1(I), b1(I)) ∀I ∈ D1.
Step 2: Convert CG(2) to MCG(2) using π(D1) obtained from Step 1. Select an assessment (σ2, b2) ∈ Ψ(MCG(2)). Set (π(I), µ(I)) = (σ2(I), b2(I)) ∀I ∈ D2.
Step n: Convert to using Sn−1 k obtained from Step 1 through Step CG(n) MCG(n) π( k=1 D ) (n-1). Select an assessment (σn, bn) ∈ Ψ(MCG(n)). Set (π(I), µ(I)) = (σn(I), bn(I)) ∀I ∈ Dn. Repeat Step n until n = S.20 At step n ∈ {1, .., S}, given MCG(n), the SE (σn, bn), is said to be the belief about the assessment (strategy and belief pro le) in MCG(n), as calculated by the player-types when moving at their respective information sets in Dn. Figure 4 below shows the construction of MCG(2), the notes below the gure specify how to solve for the LFE for our Centipede game example.
20Note 1 : In the construction of an LFE in de nition 2, we are assuming that a player type observing a longer CG correctly anticipates which one of the many possible SE was selected at each of the shorter CGs. For example, for constructing MCG(2), we need a selection from the Sequential Equilibria of CG(1). We are assuming that 02 at stage 2 and 11 at stage 1 correctly guess which one of the many possible optimal choices is chosen by 01 at stage 1. This assumption is signi cant in general, but it has no bearing on our Centipede game and Sequential Bargaining game results, as all CGs 6= Γ have a unique SE there.
20 Nature
(01, 02) (01, 12) (11, 02) (11, 12) (21, 02) (21, 12) 2 Nature 11(D ) 21
P 1S T 1 P 1S T 1 P 1∗ T 1 P 1∗ T 1 P 1 T 1 P 1 T 1 2 2 2 02(D ) 12 02(D ) 12 02(D ) 12 4, 1 4, 1 4, 1 4, 1 4, 1 4, 1
P 2∗ T 2 P 2 T 2 P 2∗ T 2 P 2 T 2 P 2∗ T 2 P 2 T 2
12, 18 2, 8 12, 18 2, 8 12, 18 2, 8 12, 18 2, 8 12, 18 2, 8 12, 18 2, 8
Figure 4: Modi ed Curtailed Game (2)
Solving for LFE. Figure 3 depicts CG(1). From CG(1) we know that in LFE, 01 chooses P 1 at the information set D1 for which CG(1) is decisive. This is irrespective of beliefs. We construct by taking 0 LFE action at 1 as Nature's move. This converts to . MCG(2) 01s D CG(2) MCG(2) MCG(2) is identical to Γ in all respects except that (i) MCG(2) curtails Γ after the second stage action and (ii) in , we take 0 LFE action at 1 as Nature's move. is decisive MCG(2) 01s D MCG(2) for information sets denoted by 2: 0 information set at stage 1, and 0 information set at D 11s 02s stage 2. In any SE of MCG(2), and therefore in LFE, 11 chooses P 1 in stage 1, and 02 chooses P 2 in stage 2. This is also irrespective of belief because in any SE of MCG(2), all player 2 types choose P 2 at stage 2. Note that we mark the SE actions at even the non-decisive information sets by underlining them. So we underline P 2 for 12 at stage 2 and T 1 for 21 at stage 1. This is because the SE actions at non-decisive information sets are needed to calculate the LFE actions and beliefs at decisive information sets. To complete the LFE strategy pro le, we need to solve for the LFE actions at the remaining information sets, D3, using the Interaction game in Figure 2. Now, Nature's initial distribution is important. Suppose Nature chooses an independent uniform distribution on player 1's types and player 2's types. At stage 3, all types of player 1 will choose
T 3, irrespective of beliefs. Thus, at stage 2, 12 will choose T 2 irrespective of beliefs. In any SE of , 0 beliefs must be derived from Nature's prior distribution using Bayes' rule. Thus, MCG(3) 21s 0 belief that his opponent is or is each. The former chooses in stage 2, while the 21s 12 02 0.5 T 2 latter chooses in stage 2, thus 0 expected payo from is 16+2 . So chooses in P 2 21s P 1 2 = 9 21 P 1 stage 1.
21 3.2 Limited Foresight Equilibrium Properties
Remark 1: The interaction game nests the underlying nite sequential move game of per- fect information. In particular, suppose an Interaction Game Γ has the following common knowledge prior distribution chosen by Nature: the combination of player-types such that none of the player-types has any limitation to their foresight has a probability of 1, i.e.
21 ρ(t1, .., tM ) = 1 if ti = (N − si)i for all i = 1, .., M. Then Γ is equivalent to Γ0, the game of perfect information that generated Γ. Therefore, in that case, the set of LFE of Γ is equal to the set of Sequential Equilibria of Γ which is identical to the set of SPNE of Γ0.
Proposition 1(a) - Existence: for every nite Interaction Game, there exists at least one Limited Foresight Equilibrium. 1(b) - Upper Hemicontinuity: Given the extensive form, , for {N,H, {I(ti)}ti∈N ,P,A} an Interaction Game, the correspondence from pairs (ρ, u) of initial probability distributions and payo pro les to the set of Limited Foresight Equilibria for the game so de ned is upper hemi-continuous.
The proof of existence and upper hemicontinuity of LFE follows from the existence and upperhemicontinuity of the Sequential Equilibrium (Kreps and Wilson (1982)). The details are given in the Appendix.
Proposition 2 and corollary tells us that calculating the SE of a given MCG(n) can be quite easy. Proposition 2 tells us that in any MCG(n), the beliefs of all types of a particu- lar player at corresponding information sets are identical. Further, as the beliefs of dierent
21For example, Palacios-Huerta and Volij (2009) may have been able to establish this condition in their experiment when expert chess players played other expert chess players in a Centipede game.
22 types of a given player are identical (that is, beliefs over the opponents' types) at correspond- ing information sets, in equilibrium, if a player-type's strategy from a given information set is strictly better than the next-best alternative, then in equilibrium, the strategies of all types of that player at the corresponding information sets are also identical. That is, all types of a particular player can be treated identically in an MCG(n) up to the case of indierence. The case of indierence doesn't arise in any MCG shorter than the Interaction game in the Bargaining game and the Centipede game applications. My conjecture is that even in the case of indierence identical treatment of dierent player-types in any MCG(n) is without loss of generality. However, I have not been able to prove this conjecture formally. To the best of my knowledge, there does not exist an application where this identical treatment of the dierent types of a particular player fails to produce a SE of some MCG(n).
Proposition 2: Consider an arbitrary modi ed curtailed game, MCG(n). Within MCG(n), consider any two information sets of and 0 such that the sequence of prior actions, ex- ti ti cluding Nature's initial move, is the same. That is, consider , and 0 such that I(ti) I(ti)
−1 −1 Seq (I(ti)) = Seq (I(ti)) = h0 ∈ H0. Further, let Nature's initial probability distribution, ρ, over player-types be pairwise independent across the types of dierent players. That is,
P rob(ti, tj|ρ) = P rob(ti|ρ).P rob(tj|ρ) for any ti ∈ Ti and tj ∈ Tj and for any i, j ∈ N0. Then for any totally mixed strategy pro le of MCG(n), denoted as n , the beliefs of and 0 σ (.) ti ti over the histories in and 0 are identical if these beliefs are calculated using Bayes' I(ti) I(ti)
n −1 −1 law given σ (.). That is, if Seq (I(ti)) = Seq (I(ti)) = h0, then
n n n 0 0 n (2) b ((t−i, h0)| ti,I(ti), σ ) = b ((t−i, h0)| ti,I(ti), σ ) ∀t−i ∈ T−i
Therefore, for any MCG(n) and for any of its Sequential Equilibria, the equilibrium be-
23 liefs of all types of player i are identical following action sequences that are identical up to Nature's initial move.
Proof sketch. We provide an intuitive explanation of why proposition 2 is true. Let P r be short for probability for the purpose of this proof. Suppose the precedent of proposition 2 holds. Then
P r((t , t , h )| σn) bn((t , h )| t ,I(t ), σn) = i −i 0 (3) −i 0 i i P [P r((t , t , h )| σn)] t−i∈T−i i −i 0
We have to show that for all , 0 , for any given , ti ti ∈ Ti t−i ∈ T−i
P r((t0 , t , h )| σn) P r((t , t , h )| σn) i −i 0 = i −i 0 (4) P [P r((t0 , t , h )| σn)] P [P r((t , t , h )| σn)] t−i∈T−i i −i 0 t−i∈T−i i −i 0
n When we calculate P r((ti, t−i, h0)| σ ), the manner in which this term is aected by ti is captured by a multiplicative term that varies based on ti, but given i's type, it doesn't vary as t−i varies (in the denominator of (4)). This is due to two main features of the structure of MCG(n). First, the independence property of ρ and, second, the construction of the information sets of MCG(n). That is, given i's type, say ti, if he is moving after a sequence of moves given by h0, he cannot distinguish among possible t−i that preceded h0, because for all t−i ∈ T−i, the history given by a dierent (t−i, h0) belong in the same information
0 set for ti. Similarly for each j 6= i, j s types' cannot distinguish among t−j for a given sequence of actions. Thus j0s types' strategy cannot specify actions which are conditional on i's type. Thus, the multiplicative term that is speci c to ti, but common to any t−i, given ti, cancels out from the numerator and denominator, leaving the same term on the RHS and LHS of (4), i.e. a term that is unaected by i's type. Details of this are given in the Appendix.
The corollary below gives us an important implication of proposition 2. Consider an arbi-
24 trary MCG(n) corresponding to an interaction game Γ and an underlying game Γ0. Consider
n n n any SE, (σ , b ), of MCG(n). If σ speci es a strategy for ti following an action sequence h0 such that it is strictly better than all alternatives. Then σn must specify the same strategy for all other types of player i at corresponding information sets.
Corollary: Suppose the conditions of proposition 2 hold. Consider an arbitrary ac-
h0 tion sequence h0 of the underlying game. Let I (ti) be the collection of information sets of t which follow after the action sequence h . Let U (σn (Ih0 (t ))|σn , bn) be the ex- i 0 ti ti i −ti pected payo of t from following the strategy σn (Ih0 (t )) over the information sets Ih0 (t ), i ti i i given the belief pro le bn, and the strategies of all other player types given by σn . If −ti
U (σn (Ih0 (t ))|σn , bn) >U (sn (Ih0 (t ))|σn , bn) for all possible strategies sn (Ih0 (t )) over ti ti i −ti ti ti i −ti ti i
h0 , then in equilibrium, the strategies of all other types of player , for example 0 , must I (ti) i ti
n h0 0 n h0 0 h0 0 be such that σ 0 (I (t ))= σ (I (ti)), where for each I(t ) ∈ I (t ), there exists a unique ti i ti i i
h0 such that −1 −1 0 . (The proof is given in the Appendix). I(ti) ∈ I (ti) Seq (I(ti)) = Seq (I(ti))
De nition 5 (lower types and higher types): consider an arbitrary information stage-K information set of . Let k : . De ne as I(ti) Γ I(ti) = {(t, (a0)k=1,...,K−1) t−i ∈ T−i} L(I(ti)) the subset of such that k if and only if for any subse- I(ti) h = (t, (a0)k=1,...,K−1) ∈ L(I(ti)) quence ˆ k of it is true that if ˆ then . Note h = (t, (a0)k=1,...,r−1) h P (h) = tj (tj + r) < (ti + K) ˆ that h is a stage r history of Γ and tj denotes player-type tj's foresight level, so (tj + r) is ˆ tj's total foresight at h, and similarly (ti +K) is ti's foresight level at h. For all h ∈ L(I(ti)), ti is said to be playing against lower opponent types, that is, at h, ti has greater total fore- sight that all of his opponents did in their respective prior moves in the sequence h. De ne
c L (I(ti)) = I(ti) − L(I(ti)).
25 Proposition 3 states a consistency condition on the LFE belief µ. It says that in any LFE (π, µ), given an information set I, the belief distribution over L(I), should be derived from the LFE strategy pro le π using Bayes' rule wherever possible.
Proposition 3: Let I be a stage-K information set of Γ, such that P (I) = ti. In any LFE , 0 belief distribution over the set of nodes , conditional on , must be derived (π, µ) tis L(I) I from the LFE strategy pro le using Bayes' rule wherever possible. That is, ∀h ∈ L(I), if
P rob(L(I) | ρK+ti ) = P rob(L(I) | π) > 0 we must have:
P rob(h | π) P rob(h | ρK+ti ) µt (h| L(I)) = = (5) i P rob(L(I) | π) P rob(L(I) | ρK+ti )
A sketch of the proof is stated here. In constructing any LFE (π, µ), when we get to step
(K+ti) of the construction, we construct MCG(K+ti) using steps 1 through (K+ti −1). We have already solved for the LFE strategies for all shorter MCGs. We consider the strategies of lower opponent types of ti at stage K as Nature's moves, which are common knowledge in MCG(K + ti). Therefore, conditional on ti moving after some sequence such that all prior moves are those of some lower opponent type, that is, conditional on ti being at L(I), the probability belief on each such individual sequence in L(I) is calculated using Nature's moves which are common knowledge in MCG(K + ti). By de nition, these Nature's moves are given by the LFE strategy pro le, π. Technical details are stated in the Appendix.
Corollary: Consider two stage- information sets and 0 of such that they occur s I(ti) I(ti) Γ
26 after the same history of actions. That is, k : and I(ti) = {((ti, t−i), (a0)k=1,...,s−1) t−i ∈ T−i} 0 0 k : . If 0 then we must have that I(ti) = {((ti, t−i), (a0)k=1,...,s−1) t−i ∈ T−i} ti < ti |L(I(ti))| ≤ 0 . Therefore, by proposition 3, the LFE conditional belief distribution of the higher |L(I(ti))| foresight-level type is accurate (satis es ) on a larger subset of 0 as compared to the (5) I(ti) subset of I(ti) for which the lower foresight-level type's conditional belief distribution is ac- curate (satis es (5)). (The proof to this corollary is given in the Appendix.)
The corollary to proposition 3 (stated above) approximately captures the ndings from Reynolds (1992). Reynolds (1992), while testing recognition of opponent's expertise among chess players, found that Higher rated players consistently made lower estimation errors (of other chess players' ELO ratings). If one proxies for foresight using experience-level or ELO ratings, then the corollary to proposition 3 approximately captures this. The reasons for only approximate similarity to Reynolds' (1992) ndings are that rst, the proxying of foresight using ELO ratings is a leap of faith; second, in an LFE, the total believed prob- ability on lower types, , need not be derived from the LFE strategy µti (L(I(ti))| I(ti)) pro le using Bayes' rule (Proposition 4). However, as proposition 3 says, conditional on , the distribution of among the various sequences of , i.e. L(I(ti)) µti (L(I(ti))| I(ti)) L(I(ti)) the distribution of among lower types, is derived from the LFE strategy µin (L(I(ti))| I(ti)) pro le using Bayes' rule. It is notable that starting from the same common knowledge belief over opponents' types, the belief of higher foresight-level types becomes more accurate (at least in the sense of proposition 3 and its corollary) after the same sequence of actions.
Proposition 4: If the total foresight of ti at the stage-s information set I(ti) is less than
S, i.e. s + ti < S, then his belief distribution conditional on the histories in I(ti) need not be derived from the LFE strategy pro le using Bayes' rule. Thus, it need not be true that
P rob(h| π) that µt (h| I(ti)) = ∀ h ∈ I(ti) and for all information sets I(ti) of Γ . i P rob(I(ti)| π)
27 Proof by counterexample. (To be written afresh. See Limited Foresight and Learning Equilibrium by Rampal (2016) available at https://sites.google.com/site/jeevantrampale- con/Research for a proof of this).
Proposition 4 follows because at the information sets of MCG(s+ti), the strategy pro le
(s+ti) for MCG(s + ti), σ , may stipulate dierent optimal actions compared to the LFE strategy pro le, π. Although σ(s+ti) can be dierent from the LFE pro le ,π, it provides an optimal strategy with respect to MCG(s + ti) for each player type in MCG(s + ti) given the strategy and belief pro les (σ(s+n), b(s+n)). Proposition 5 (below) tells us that if a low foresight type observes a sequence of moves that cannot occur when playing against lower opponent types, then he discovers that he is playing against some higher type, and must use his total foresight at that move to opti- mize. At any information set, I(ti), where a limited foresight player-type ti moves, there is a certain subset of nodes, L(I(ti)) ⊂ I(ti), which represent the cases where ti is playing against lower opponents' types who, at all preceding moves leading to that information set, had a strictly lower total foresight than ti does at h. Proposition 5 (below) re ects the fact that ti knows these lower opponents' types' prior moves coming into these nodes of L(I(ti)). If ti knows that the moves of these lower opponents' types' imply a zero probability of reaching any node in L(I(ti)), and yet nds himself at the information set I(ti), then he knows that he is at a node of I(ti) where at least one of the opponent-type was not a lower type at some preceding move. In the two player case it means that one knows that one's opponent had a higher total foresight at some preceding move. Recognition of the higher opponent type implies that the LFE actions of the lower types don't matter for the calculation of the sequentially rational action at I(ti); ti must use his total foresight to optimize.
28 Proposition 5: If in the construction of an LFE, Nature's moves in MCG(s+ti), denoted
s+ti by ρ , imply that the probability of reaching I(ti), a stage-s information set, via only
Nature's moves is 0, then the LFE belief of ti, conditional on I(ti), must put probability 1 on those nodes of I(ti) where at some preceding node, the player type moving there had total foresight at least (s + ti) . That is, for all ti ∈ N, for all I(ti) ∈ I(ti):
c P rob([L(I(ti))] )| π) s+ti c (6) [P rob(L(I(ti)) | ρ ) = 0] =⇒ µti ([L(I(ti))] | I(ti)) = = 1 P rob(I(ti)| π)
Proof: proposition 5 follows from the stepwise de nition of LFE. By the LFE de nition, all the nodes preceding the nodes in L(I(ti)) have Nature as the player moving there and
s+ti s+ti the actions taken by Nature are given by ρ . ρ is common knowledge in MCG(s + ti) and hence also known to ti at I(ti). Thus, the probability of reaching I(ti) via only Nature's
s+ti moves, can be calculated using ρ by ti at I(ti). Thus,
s+ti P rob(L(I(ti))| ρ ) µti (L(I(ti)) | I(ti)) = µti (I(ti))
Therefore, if s+ti , then , further, as c P rob(I(ti))| ρ ) = 0 µti (L(I(ti)) | I(ti)) = 0 µti ([L(I(ti))] | I(ti)) , we must have that c .Q.E.D. +µti (L(I(ti)) | I(ti)) = 1 [µin ([L(I(ti))] | I(ti)) = 1]
0 Remark 2: Suppose ti moves at two information sets I(ti) and I (ti), which occur at
0 stage s and s , respectively, of Γ. By the construction of LFE, in step (s + ti), ti at stage s knows ρ(s+ti), and therefore he knows the LFE action choices of player-types at all the
29 decisive information sets of through , S(s+ti−1) n. If 0 then MCG(1) MCG(s + ti − 1) n=1 D s > s
0 0 0 (s +ti) 0 ti at s , by step (s + ti) of the construction of LFE, knows ρ , and therefore ti at s
0 knows the LFE action choices of player-types at more information sets S(s +ti−1) n of n=1 D Γ 0 S(s +ti−1) n S(s+ti−1) n. n=1 D ⊃ n=1 D
Remark 2 approximately mirrors another nding from Reynolds (1992), and the nding from Rampal (2017). Rampal (2017) found that the more moves of the opponent observed by an expert race game player, the better his guess about the opponent's experience level. In the same token, Reynolds (1992) found that the estimation error decreased as a function of number of moves revealed. Remark 2 suggests that in an LFE this can happen, as at a higher stage number, the same player type has a higher total foresight and hence observes a longer Curtailed Game. Thus, LFE actions are given as Nature's move for a larger subset of the set of information sets of the Interaction Game.
4 Applications
A key aim for developing the LFE apparatus is to obtain general applicability in solving various existing puzzles observed in the experimental data collected on perfect information games. In this section we apply the LFE model to the Centipede game introduced by Rosenthal (1981) and the Sequential Bargaining game analyzed by Rubinstein (1982) and Ståhl (1973).
4.1 Sequential Bargaining
The Sequential Bargaining game (Rubinstein (1982) and Ståhl (1973)) has been studied extensively in the literature (c.f. Binmore et al (1985), Neelin et al (1988), Guth and Tietz (1987,1990), Ochs and Roth (1989), Johnson et al (2002), and Binmore et al (2002)). The
30 game consists of two players bargaining over a pie of size X over multiple periods. In each period one player makes a proposal on how to split the pie, and the other player accepts or rejects this proposal. If a proposal is accepted then the game ends and that proposal is implemented. If a proposal is rejected then the game proceeds to the next period where the player who rejected the last proposal now makes an oer but from a smaller pie as the pie gets multiplied by a common discount factor, δ ∈ [0, 1]. In the nite period case, if no proposal is accepted, then after a rejection in the last period, both players get 0 payo. The
SPNE prediction is that in a K period bargaining game, when K is odd, the rst proposal which oers the rst mover/proposer (1−δK−1) K−1 will be accepted. X[(1 − δ) 1−δ2 + δ ] Four stylized data trends, which are incongruent to the SPNE outcomes, have emerged in the experimental study of the Sequential Bargaining game. First, a tendency for rst oers proposing equal split (Guth and Tietz (1987); Ochs and Roth (1989)) or oering the second round pie (Neelin et al (1988)) to the second mover. Second, oers made in the rst period are often rejected (Ochs and Roth (1989)). Third, and perhaps the most surprising nding is that the rst period oers are very often succeeded by disadvantageous counteroers (Ochs and Roth (1989) found that 81 percent of counteroers were disadvantageous). Fourth, subgame consistency is violated in that observed outcomes of a subgame tested as a separate game are dierent from the outcomes of this subgame when it is the strict subgame of a game (Binmore et al (2002)). In this subsection, we show that we can rationalize all these stylized data facts simultaneously by just utilizing the general model of limited foresight and uncertainty about the opponent's foresight that we developed earlier. In particular, our rationalization does not use altruistic preferences, or preferences speci c to the Bargaining game.
We consider the three period bargaining game with δ = 0.6 as Γ0. These speci cations are used by Ochs and Roth (1989) in one of their treatments. Neelin et al (1988) and Johnson et al (2002) use δ = 0.5, which doesn't change the features of the LFE outcome we discuss below. We make the initial size of pie 1000 for simplicity. We convert Γ0 into the Interaction
31 Game, Γ, and present the features of its LFE. The Figure 5 below depicts the Curtailed payos associated with Γ0, without showing the informational uncertainty.
Figure 5: Sequential Bargaining Game and Associated Curtailed Payos Without Uncertainty The gure shows curtailed payo pro les being calculated using the (min+max)/2 method. The curtailed payos are depicted in blue above the game. The pies are 1000, 600 and 360 in period 1,
2 and 3 respectively. xi is player 1's oer (to himself) in period i. y2 is player 2's oer ( to player 1) in period 2. R implies reject and A implies accept.
As the three period bargaining game given in the Figure 5 above has 6 stages, we have six player-1 types (01, 11, 21, 31, 41, and 51) and ve player-2 types (02, 12, 22, 32, 42). We assume independent uniform distributions on both players' types. The LFE strategies for this uniform case are calculated in the Appendix and detailed in Table 1 there. The following outcomes observed by the studies on the Bargaining game (mentioned in brackets) are observed in the LFE that we detail in the Appendix.
1. First round oer rejection (c.f. Ochs and Roth (1989)): 01 overestimates his bargaining position and thus demands the whole rst period pie. This demand is rejected by all
player-2 types. The oer of 11 and 21 is rejected by 22 and 32. This occurs because
22 and 32 fail to take into account that player-1 has absolute bargaining power in the last period and that the pie will shrink in the next period when they have to make a counterproposal.
2. First oers with near equal split or an oer equal to the second round pie (c.f. Neelin
et al (1988); Guth and Tietz (1987); Ochs and Roth (1989)): 31, 41, and 51 propose
32 (580, 420) in the rst period. 31, and 41 choose this proposal because they cannot forsee
that they will have the bargaining advantage in the last (third) period, 51 chooses this
proposal because he gets immediate acceptance with this generous oer. If 51 were to make a higher oer, his oer would be rejected by the limited foresight types of player-2 who fail to forsee player-1's absolute bargaining power in the last period.
3. Disadvantageous counter proposals (c.f. Ochs and Roth (1989)): 11 and 21 make a
proposal of (700, 300) in the rst period. However, 22 and 32 reject anything that gives them less than 420 because, due to their limited foresight, they think that all player-1
types will accept a proposal of (180, 420) in the second period. However, 32 becomes rational in period 2 and observes the absolute bargaining power of his opponent in
the last period. Thus, it is sequentially rational for 32 to make a disadvantageous counterproposal of (360, 240). A theoretical prediction of the LFE model is that this feature should disappear if we change the extensive form and make player-2 think about the acceptance/rejection decision simultaneously with the counterproposal decision. Thus, one should take great care in matching the speci cation of moves in the game to the foresight of the players.
4. Subgame consistency violation (c.f. Binmore et al (2002)): Consider the 2-period
Bargaining game with the starting pie of 600 being tested separately and its data being compared to the data generated from the last two periods of a 3-period Bargaining game. Binmore et al (2002) nd that the results of the former do not match the data generated from the latter. According to LFE, this is to be expected if these seemingly perfect information games are in fact Interaction games. This because in the 3-period game, the outcome of the last two periods depends on what happened
in the rst period. For example if the rst proposal was (1000, 0), then player-1's
type is 01, and if the rst proposal was (700, 300) then player-1's type can be 11 or 21 with equal probability. These dierent player-1 types have dierent optimal choices
33 in the third period. Further player-2's types 32 and 42 update about their opponent's type based on the rst period proposal and adjust their optimal actions in the second period. However, if two players are beginning a two-period Bargaining game, then their optimal choices only depend on their prior belief about the opponent's foresight, which may well be dierent from their updated belief after observing the opponent's choice in the rst period of a 3-period Bargaining game.
Thus the LFLE concept provides us several channels to explain several qualitative features of the data on Sequential Bargaining experiments. Fitting experimental data using this model is left as future work.
4.2 The Centipede Game
The Centipede game describes a situation in which two players alternately decide whether to take or pass an increasing pile of money. Consider an S-staged Centipede game. First, player 1 decides whether to take or pass a pile of money; if the player moving at stage i decides to take at stage i then he gets ai, the larger share of the existing pile of money, ai + bi. If that player passes, the pile of money grows and ai + bi < ai+1 + bi+1. If a player passes, but his opponent takes in the next stage, he gets a payo bi+1 < ai. However if his opponent passes too, then the pile grows again and the player has a chance to take again and achieve a higher payo ai+2 > ai. bS+1 denotes the payo of the player moving at stage S, if he chooses pass at stage S. His opponent gets aS+1 > bS+1. The unique SPNE prediction is that the rst player should take in the very rst stage, regardless of the number of stages that the pile can be passed and grown. The logic is that in the last stage, as aS > bS+1, the player moving there should take ; but given this, one should take in the second-last stage, and this optimality of taking given one's opponent is going to take in the next stage continues inexorably backwards, and leads to the SPNE prediction: take in the rst stage. This is highly unintuitive and various experiments, eg. McKelvey and Palfrey (1992, 1998) reject the SPNE prediction.
34 Figure 6: The Six Staged Centipede Game
Consider an S-staged Centipede game as Γ0. We restrict our analysis to the Centipede games with the following payo structure. De nition 3: An S-staged Centipede game is said to have the payo structure P if for all
bi+1+ai+2 ai−bi+1 1 i ∈ {1, .., S + 1}: (i) bi < bi+1 < ai < bi+3 < ai+2 (ii) ai < (iii) = ηi < . 2 ai+2−bi+1 3 Consider the six staged Centipede game used by McKelvey and Palfrey (1992) in gure 6.
This also has the payo structure P with 1 for all . If a term, for example , does not ηi = 7 i bi+3 exist then any condition on that term is satis ed vacuously. bi+1 < ai < ai+2 follows from Γ0 being a Centipede game. Condition (iii) of de nition 3 will be used in proving proposition 7 below. An S-staged perfect information Centipede game Γ0 generates an Interaction Game Γ with the player set N = {01, 11, ..., (S −1)1, 02, 12, ..., (S −2)2}. The following proposition says that given a certain form of initial probability distribution on foresight types, even with arbitrary positive total probability on limited foresight types, all LFLE outcomes entail pass being played with strictly positive probability by all foresight types until the end stages of a Centipede game. This result re ects the fact that rational the rational type player pretends to be low foresight types and attain a higher payo by passing because his opponent cannot tell if he is rational or a limited foresight type, and thus the rational opponent passes with positive probability too.
Proposition 7: Consider an S-staged Centipede game Γ0 with payo structure P. Γ0 gen- erates a S-staged Interaction Game Γ. Let ρ, the probability distribution on N be such that
35 P rob(j1) = P rob(k2) = q ∈ [0, 1], ∀ j = 0, 1, ..., S − 2 and ∀ k = 0, 1, 2, ...., S − 3, and
P rob((S − 1)1) = 1 − (S − 1)q, and P rob((S − 2)2) = 1 − (S − 2)q. Further suppose the distribution on 10s types is independent of the distribution on 20s types. For all q > 0 such that PS−1 PS−2 , in any LFLE of , all types of both players pass j=0 P rob(j1) = k=0 P rob(k2) = 1 Γ with strictly positive probability from stage 1 through stage (S-3).
Proof sketch: rst we show that any limited foresight type, at any stage at which his total foresight is strictly less than S, plays pass with probability 1. The proof proceeds to show that this fact implies that if the rational player-types all stop at a particular stage, say s, between 1 and (S − 3), then in the next stage, the rational player-types face limited- foresight opponent types who played pass at stage s, and out of which only one will turn rational in stage (s + 2). Thus, if the rational player-types all stop at stage s, then in stage (s + 1), the rational player-types know that their opponent will pass with a high probability in stage (s + 2), which implies that all player-types pass in stage (s + 1), but that means that stopping at stage s is not sequentially rational for the rational player types at stage s, and therefore not an LFE. The technical details are given in the Appendix.
This analysis is almost parallel to the McKelvey and Palfrey (1992) model without the errors in actions, heterogeneous beliefs and learning components. Both these analyses are in the same vein as the reputation literature of Kreps, Wilson, Milgrom and Roberts (1982).
In particular, if we have a ρ such that P rob(01) = P rob(02) = 1 − q and P rob((S − 1)1) =
P rob((S − 2)2) = q then we can use McKelvey and Palfrey (1992) to characterize the unique LFLE. The only dierence would be that we would have to replace S by S − 1 in their analysis as their altruist type (corresponding to 01, 02), who occurs with probability (1 − q) chooses pass in all stages , while even the lowest foresight types in our analysis, 01 and 02, take in the Sth stage.
36 5 Conclusion
This paper de nes the Limited Foresight Equilibrium (LFE). The LFE is de ned for general applicability in the class of nite sequential move games with perfect information. In seeking to make more intuitive and experimentally justi able predictions, we model the case where players are interested in maximizing own payo, but each player possesses one of dierent levels of foresight. Further, players are uncertain about their opponents' foresight. The LFE model nests the perfect information case. We prove the existence, upperhemicontinuity and other properties of LFE that seeks to capture the dynamics of real life nite sequential move games with seemingly perfect information. These properties are: (a) The higher the foresight-level of a player, the better he can estimate his opponents' foresight. (b) The more moves any player-type observes, the better he becomes at guessing the opponent's foresight level. (c) If a low foresight type is surprised by a sequence of moves impossible against lower types, he discovers that he is playing against some higher type, and must use his total foresight at that move to optimize. From (a), (b), and (c) we obtain: (d) The high foresight type must choose between revealing his type or pretending to be a low type. We show the applicability of LFLE in two existing puzzles in the class of nite, two player alternate move games, namely, the Centipede Game and the Sequential Bargaining game. In the Centipede Game, LFLE unleashes reputation eects, as in Kreps, Wilson, Milgrom and Roberts (1982), and McKelvey and Palfrey (1992), which lead to cooperative behavior even among rational players. In the Sequential Bargaining application, these features of the LFE help rationalize the disparate ndings from the study of bargaining: namely, LFE produces outcomes that show (i) rst round oer rejection (ii) rst round oer of near equal split (iii) disadvantageous counter proposals (iv) subgame inconsistency. These LFE results for Sequential Bargaining are parallel to several qualitative results in dierent experimental studies on bargaining.
37 6 Appendix
Proof of proposition 1(a) (Existence of LFE): Consider an arbitrary nite Interaction
Game Γ. The CG(1) derived from Γ is also nite. Due to proposition 1 of Kreps and Wilson (1982), there exists a SE of CG(1). We can select an arbitrary SE(1) of CG(1) to construct MCG(2). MCG(2) is also nite. Thus the SE(2) of MCG(2) also exists. Proceeding thus, given the existence of SE of each of CG(1),MCG(2),...,MCG(n − 1), we can construct MCG(n) in step n of de nition 2. As MCG(n) is nite, there exists a SE of MCG(n). As this holds for n = 2, ..., N each of the steps in de nition 2 can be carried out as de ned, and thus there exists at least one LFE of Γ
An inductive argument for Upperhemicontinuity of LFE Consider an arbitrary -staged Interaction Game . For a given extensive form, , let S Γ {N,H, {I(ti)}ti∈N ,P,A} the correspondence f : ∆T × RN =⇒ Π × M be the set valued function, mapping initial conditions and payos ,(ρ, u), to the set containing all associated LFE assessments. An element of the set f(ρ, u) is an LFE, denoted as (π, µ). Fix a sequence (ρk, uk) → (ρ, u) and an associated sequence (πk, µk) ∈ f(ρk, uk), such that (πk, µk) → (π, µ). To show upperhemicontinuity, we need to show that (π, µ) ∈ f(ρ, u).
Given an arbitrary extensive form 0 0 0 0 0 , let N 0 {N ,H , {I (ti)}ti∈N ,P ,A } Ψ : ∆T ×R =⇒ Σ×B be the Upper Hemi Continuous (UHC) correspondence mapping initial conditions and payos, (ρ0, u0), to the set Ψ(ρ0, u0), which contains all the sequential equilibrium strategies and beliefs, (σ, b), of the game so de ned. Let π(Hˆ ), µ(Hˆ ) denote the vectors π and µ restricted to the coordinates correspond- ing to the information sets contained in Hˆ . Let f(ρ, u)(Hˆ ) also represent each element of f(ρ, u) restricted to Hˆ . We prove upperhemicontinuity by induction. Step 1: we show
1 1 1 that (π(D ), µ(D )) ∈ f(ρ, u)(D ). Consider CG(1) = MCG(1). Corresponding to (ρk, uk) we have 1 1 for each element of the sequence . The superscript denotes the (ρk, uk) k = 1, 2, ... length of the . The construction of 1 using the curtail and min+max method is described CG uk 2
38 in section 2. As the function which maps a nite set of real numbers to their min+max is a 2 continuous function, implies 1 . Also, 1 , so implies uk → u uk → u1 ρk = ρk (ρk, uk) → (ρ, u) that 1 1 1 1 . Now note that for each in the sequence, 1 1 (ρk, uk) → (ρ , u ) k (πk(D ), µk(D )) = 1 1 1 1 , and 1 1 1 1 1 1 1 . We know that is UHC. Thus (σk(D ), bk(D )) (σk(D ), bk(D )) ∈ Ψ(ρk, uk)(D ) Ψ(.) if 1 1 1 1 1 1 1 1 then 1 1 1 1 1 1 1 . Given that (σk(D ), bk(D )) → (σ (D ), b (D )) (σ (D ), b (D )) ∈ Ψ(ρ , u )(D ) 1 1 1 1 , and given that 1 1 1 1 1 1 (πk(D ), µk(D )) → (π(D ), µ(D )) (πk(D ), µk(D )) = (σk(D ), bk(D )) → (σ1(D1), b1(D1)) it follows from the uniqueness of a limit that (σ1(D1), b1(D1)) = (π(D1), µ(D1)) ∈ Ψ(ρ1, u1)(D1) ⊂ f(ρ, µ)(D1). Therefore (π(D1), µ(D1)) ∈ f(ρ, µ)(D1).
Step 2: Consider , where . Let Si=n−1 i Si=n−1 i MCG(n) n ∈ {2, .., N} (π( i=1 D ), µ( i=1 D )) ∈ Si=n−1 i . We will show that Si=n i Si=n i Si=n i . Given f(ρ, u)( i=1 D ) (π( i=1 D ), µ( i=1 D )) ∈ f(ρ, u)( i=1 D ) step 1, this will complete the proof.
Corresponding to we have n n for each . Using Si=n−1 i (ρk, uk) (ρk , uk ) k = 1, 2, ... πk( i=1 D ) and we generate n as detailed in section 2. By continuity, n n. As Si=n−1 i ρk ρk uk → u πk( i=1 D ) → Si=n−1 i by assumption, thus: (i) n n n n and (ii) it will su ce to show π( i=1 D ) (ρk , uk ) → (ρ , u ) n n n . Now note that for each , n n n n n n , (π(D ), µ(D )) ∈ f(ρ, u)(D ) k (πk(D ), µk(D )) = (σk (D ), bk (D )) and n n n n n n n . We know that is UHC. Thus, if n n n n (σk (D ), bk (D )) ∈ Ψ(ρk , uk )(D ) Ψ(.) (σk (D ), bk (D )) →
n n n n n n n n n n n n n (σ (D ), b (D )), then (σ (D ), b (D )) ∈ Ψ(ρ , u )(D ). Given that (πk(D ), µk(D )) → n n , and given that n n n n n n n n n n , (π(D ), µ(D )) (πk(D ), µk(D )) = (σk (D ), bk (D )) → (σ (D ), b (D )) by the uniqueness of a limit, it follows that (σn(Dn), bn(Dn)) = (π(Dn), µ(Dn)) ∈ Ψ(ρn, un)(Dn) ⊂ f(ρ, µ)(Dn). Therefore, (π(Dn), µ(Dn)) ∈ f(ρ, u)(Dn). Q.E.D.
Proof of proposition 2 Let P r be short for probability for the purpose of this proof. Suppose the precedent of proposition 2 holds. Then
P r((t , t , h )| σn) bn((t , h )| t ,I(t ), σn) = i −i 0 (7) −i 0 i i P [P r((t , t , h )| σn)] t−i∈T−i i −i 0
We have to show that
39 P r((t0 , t , h )| σn) P r((t , t , h )| σn) i −i 0 = i −i 0 (8) P [P r((t0 , t , h )| σn)] P [P r((t , t , h )| σn)] t−i∈T−i i −i 0 t−i∈T−i i −i 0
n When we calculate P r((ti, t−i, h0)| σ ), the manner in which this term is aected by ti is captured by a multiplicative term that varies based on , but given 's type, or 0 , it doesn't ti i ti ti vary as t−i varies (in the denominator of (4)). Thus, the multiplicative term that is speci c to ti, but common to any t−i, given ti, cancels out from the numerator and denominator, leaving a term that is unaected by i's type.
Without loss of generality, let k . Let the information set containing the h0 = (a0)k=1,...,K history k be denoted as k . Using the independence of we get the (t, (a0)k=1,...,r) I(t, (a0)k=1,...,r) ρ following.
n n 1 n 2 1 n K k P r((ti, t−i, h0)| σ ) = P r(ti|ρ).P r(t−i|ρ).σ (a0|I(t)).σ (a0|I(t, a0))...σ (a0 |I(t, (a0)k=1,...,K−1)) (9)
De ne a subsequence or a subhistory of as k , such that (t, h0) (t, (a0)k=1,...,r) r ∈ {1, .., (K − and there exists a unique sequence of actions k such that k k 1)} (a0)k=r+1,...,K (t, (a0)k=1,...,r, (a0)k=r+1,...,K ) =
(t, h0). For any (ti, t−i, h0) such that t−i ∈ T−i, let R(ti) be the collection of natural num- bers such that the player-type moving at the subhistory k r(ti) ∈ {1, .., K} (t, (a0)k=1,...,r(ti))
c of (t, h0) is ti. Further, let R (ti) = {1, .., K} − R(ti). The set R(ti) does not depend on t−i.
It only depends on the h0 component of (t, h0). It is also worth noting that by construc- tion, for any 0 , k 0 k because by t−i, t−i ∈ T−i I((ti, t−i), (a0)k=1,...,r(ti)) = I((ti, t−i), (a0)k=1,...,r(ti)) construction, information sets in MCG(n) only re ect the uncertainty about the opponents' types, that is, uncertainty about t−i. Therefore, given an action sequence h0, changing the pro le of opponents' types preceding h0 leaves us in the same information set. We can rewrite (5), we get that
40 n n s+1 k c P r((ti , t−i, h0)| σ ) = P r(t−i|ρ)Πs∈R (ti)[σ (a0 | I(t, (a0)k=1,...,s))] n r+1 k (10) × P r(ti|ρ)Πr∈R(ti)[σ (a0 | I(t, (a0)k=1,...,r))]
P r((t , t , h )| σn) i −i 0 = P [P r((t , t , h )| σn)] t−i∈T−i i −i 0 n s+1 k P r(t |ρ)Π c [σ (a | I(t, (a ) ))] −i s∈R (ti) 0 0 k=1,...,s (11) P n s+1 k P r(t |ρ)Π c [σ (a | I(t, (a ) ))] t−i∈T−i −i s∈R (ti) 0 0 k=1,...,s
The proof will be complete if we show that the RHS of (7) does not depend on the type of player , . Consider some c . Suppose n k . Then i ti s ∈ R (ti) P (t, (a0)k=1,...,s−1) = tj k 0 k for any 0 , by the I((ti, tj, t−(i,j)), (a0)k=1,...,s)) = I((ti, tj, t−(i,j)), (a0)k=1,...,s)) ti, ti ∈ Ti construction of information sets in MCG(n). Therefore, by the de nition of an informa- tion set, n s+1 k n s+1 0 k . σ (a0 | I((ti, tj, t−(i,j)), (a0)k=1,...,s))] = σ (a0 | I((ti, tj, t−(i,j)), (a0)k=1,...,s))] Thus, the RHS of (7) does not depend on , and remains constant across 0 . Q.E.D. ti ti, ti ∈ Ti
Proof of corollary to proposition 2 By proposition 2, if −1 −1 0 , Seq (I(ti)) = Seq (I(ti)) then n n 0 . Further, for corresponding information sets, other players' ( ) b (I(ti)) = b (I(ti)) j 6= i types cannot choose dierent actions for dierent types of player . Thus, and 0 face the i ti ti
n h n n h 0 n n same strategy pro le . Therefore 0 0 0 ((σ )tj ∈Tj )j6=i Uti (s(I (ti))|σ , b ) = Ut (s(I (t ))|σ 0 , b ) tj −ti i i −ti
n h0 0 n n n h0 0 n n for all strategies s(.). Then as Ut0 (σ 0 (I (t ))|σ 0 , b ) >Ut0 (s 0 (I (t ))|σ 0 , b ) for all pos- i ti i −ti i ti i −ti
n h0 0 h0 0 n sible strategies s 0 (I (t )) over I (t ) and as σ is a SE strategy pro le, we must have that ti i i
n h0 0 n h0 −1 h0 −1 0 σ 0 (I (t )) = σ (I (ti)) when Seq (I (ti)) = Seq (I(t )). Q.E.D. ti i ti i
Proof of proposition 3 Consider an arbitrary LFE, , of . To calculate (π, µ) Γ µti (h| L(I)) within (π, µ), we need to complete steps 1 through (ti +K −1) of the construction of the LFE
(π, µ). In step (K + ti), we construct MCG(K + ti). Consider an arbitrary h ∈ L(I). Let, without loss of generality, be of the form k . In constructing h h = (t, (a0)k=1,...,K−1) MCG(ti +
41 K), for any such h ∈ L(I), using the de nition of L(I), it follows that for all subsequences of
of the form ˆ k such that , (K+ti) ˆ because h h = (t, (t, (a0)k=1,...,r)) r ≤ (K − 1) P (h) = Nature
ˆ SK+ti−1 n. We know SK+ti−1 n by steps 1 though of the construction h ∈ n=1 D π( n=1 D ) (K + ti − 1) of an LFE. Thus, in constructing MCG(s + n), we set ρK+ti (hˆ) = π(hˆ) for all subsequences
K+t −1 K+t −1 ˆ of each . So S i n K+ti S i n , and therefore is h h ∈ L(I) π( n=1 D ) = ρ ( n=1 D ) µin (h| L(I)) calculated using ρK+ti and Bayes' rule wherever P rob(L(I) | ρK+ti ) = P rob(L(I) | π) > 0. Q.E.D.
Proof of corollary to proposition 3 Suppose k , h = ((ti, t−i), (a0)k=1,...,s−1) ∈ L(I(ti)) then we will show that 0 0 k 0 to complete the proof. All h = ((ti, t−i), (a0)k=1,...,s−1) ∈ L(I(ti)) subsequences of can be written as k for some . Fix an arbitrary h ((ti, t−i), (a0)k=1,...,r−1) r < s subsequence of of the form ˆ k , it must be true that if ˆ h h = ((ti, t−i), (a0)k=1,...,r−1) P (h) = tj then r + tj < s + ti. By the construction of Γ using the Seq(.) function, we must have that for the same , the subsequence of 0 given by ˆ0 0 k is such that r h h = ((ti, t−i), (a0)k=1,...,r−1) ˆ0 . Given that 0 , we must have that 0 . Repeating this P (h ) = tj ti < ti r + tj < s + ti < s + ti argument for every , we have that for every subsequence ˆ0 0 k , r < s h = ((ti, t−i), (a0)k=1,...,r−1) for some , if ˆ0 then 0 and therefore that 0 0 . Q.E.D. r < s P (h ) = tk r + tk < r + ti h ∈ L(I(ti))
Proof of the proposition on the Centipede Game Lemma 1: Consider any stage k
Lemma 1 follows straightforwardly due to the properties of payo structure P.22For ex- ample, curtailing gure 6 at stage 3, we get (x4, y4) = (132, 80), the minimum of which, 80, 22Lemma 1 also holds with the mean of stagewise means rule followed in an earlier version of this paper. The min+max rule is only signi cant for lemma 1. Therefore it follows that proposition 7 also holds with the 2 mean of stagewise means rule.
42 is higher than the maximum number in {(4,1), (2,8), (16,4)}, 16. Due to lemma 1, in any
Curtailed Game not equal to Γ (shorter than Γ), the highest payo for both players occurs after pass at the last stage. So, irrespective of ρ, Nature's initial probability distribution on N, for all CG(1),MCG(2), ...., MCG(S − 1), there is a unique sequential equilibrium consisting of all player types playing pass with probability 1, and estimating that all other player types do the same. Thus, any limited foresight type, at any stage where he cannot observe Γ, chooses pass with probability 1. This implies that in MCG(S), according to ρS, any Nature's move, at any non initial node, speci es the pure action: pass.
The decisive information sets of MCG(S) are those where the player type moving there is rational (has full total foresight) and thus can observe MCG(S). Now, we analyze the condition for the rational player types, i.e. player types who can observe Γ, or MCG(S) to pass with strictly positive probability from stages 1 through (S − 3). We will prove this by contradiction. That is, we will show that it cannot be a SE of MCG(S), and hence cannot be an LFE, for all the player types who turn rational (attain total foresight at least K) at some stage of MCG(S) to choose strategies that imply that all rational types choose take with probability 1 at a stage before stage (S − 2). Using the homogeneity of beliefs as detailed in proposition 2, let ri denote the identical probability belief of every rational player type at stage i that at stage (i + 1) the opponent will be Nature playing (and, by lemma 1, choosing pass) on behalf of a limited foresight opponent type, conditional on play reaching stage i.
Let pi denote the identical probability put on pass by every rational player type at stage i. To show the contradiction, we only need to show that in any SE of MCG(S), it cannot be the case that pi = 0, where i = 1, ..., (N − 3). Lemma 2: For any sequential equilibrium (σS, bS) of MCG(S) : (a) If σS implies that
S S pi = 1, then σ must imply that pj = 1, for j ≤ i, where i = 1, ..., S. (b) If b is such that
S at stage i, ri > ηi, then σ must imply that pi = 1, for i = 1, ..., S − 2.
S S Proof: (a) Let (σ , b ) imply that pi = 1. That is, all rational player types pass with probability 1 at stage i. Then, sequential rationality implies that according to σS, irrespective
43 of beliefs, pi−1 = 1 because any rational type's choice to pass at stage (i − 1) is going to be reciprocated by pass with probability 1 at stage i. Therefore, the payo from pass at stage (i − 1) is at least ai+1. And ai+1 > ai−1 , where ai−1 is the payo from take at stage
S (i − 1). Similarly, pi−1 = 1 implies that due to sequential rationality of σ , we must have that pi−2 = 1, and so on for all j ≤ i.
(b) Let i ∈ {1, .., (S − 2)}. Let vi be the value to the rational types moving at stage i given sequential equilibrium play from stage i on in the MCG(S). By sequential rationality
S of σ , it follows that vi ≥ ai. The payo from take at stage i is ai, the expected payo from pass at stage i is at least rivi+2 + [1 − ri]bi+1. This is because, by lemma 1, Nature chooses pass with probability 1 at stage (i + 1). If ri > ηi then the expected payo from pass is at least rivi+2 + [1 − ri]bi+1 ≥ riai+2 + [1 − ri]bi+1 > ηiai+2 + [1 − ηi]bi+1 = ai, where the last
S equality follows by the de nition of ηi. So, by sequential rationality of σ , pi = 1. Now we can prove proposition 7 for rational player types by showing that for any SE of
S S S MCG(S), (σ , b ), it cannot be the case that σ implies pi = 0 at some i ≤ S − 3. Suppose
S σ implies pi = 0 at some i ≤ S − 3. Then, because two equiprobable limited foresight types being represented by Nature in stage turn rational in stage , S−i−2 1 i i + 2 ri+1 = S−i ≥ 3 > 1 . So, by lemma 2(b), we have . But then lemma 2 (a) implies , a ηi+1 = 7 pi+1 = 1 pi = 1 contradiction. Q.E.D.
LFE calculation for the 3 period bargaining game We assume the prior to be such that 1 for , and independently, 1 for P rob(t1) = 6 t1 ∈ {01, 11, 21, 31, 41, 51} P rob(t2) = 5 t2 ∈
{02, 12, 22, 32, 42}. Let x1 (respectively x3) denote the rst mover's (player 1)'s, denoted as P1, demand for himself in the rst stage ( fth stage), when the period number is one (three) and the size of pie is 1000 (respectively 360). Thus, (1000 − x1) (respectively (360 − x3)) is the share of the rst stage ( fth stage) pie oered to player 2, denoted P2. y2 denotes P2's oer to player 1, in the third stage, when the period number is two and the size of pie is
600. Thus (600 − y2) is the share of the third stage pie demanded by P2 for himself. We
44 summarize the LFE strategies in table 1. X1 (respectively X3) denotes the maximum share of P1 out of the rst (third) period pie, such that (1000 − X1) (respectively (360 − X3)) is acceptable to P2. Y2 denotes the minimum share of the second period pie oered by P2 to P1, such that it is acceptable to P1. As per the de nition of LFE, we construct the LFLE starting with the SE of CG(1). In what follows, we specify the SE and LFE beliefs only when needed to determine optimal actions.
23 Step 1: in the unique SE(1) of CG(1), all P1 types choose x1 = 1000, and believe the
1 prior distribution on P2 types. As D consists of only 01's move at stage I, the LFLE action of 01 at stage I is x1 = 1000.
Step 2: x 01's move at stage I as Nature's move in CG(2) to generate MCG(2). In the unique SE(2) of MCG(2), all P2 types at stage II accept if x1 ≤ 700, regardless of belief on P1 types. Thus, in SE(2), all P1 types other than 01 choose x1 = 700, regardless of belief. We note LFE actions at 2, which contains 0 information set at stage I and 0 D 11s 02s information set at stage II.
Step 3: x the LFE actions at D1,D2 as Nature's moves to convert CG(2) to MCG(3).
In the unique SE(3) of MCG(3), all P2 types at stage III choose y2 = 0, irrespective of belief. Thus, the stage II SE(3) action for all P2 types is to accept if x1 ≤ 700, regardless of belief on P1 types. Thus, in SE(3), all P1 types oer x1 = 700, regardless of beliefs. We note the LFE actions at 3, which contains 0 information set at stage I, 0 information D 21s 12s set at stage II, and 0 information set at stage III. 02s Step 4: x the LFE actions at S3 n, solved above, as Nature's moves to convert n=1 D CG(4) to MCG(4). In the unique SE(4) of MCG(4), all P1 types at stage IV accept if y2 ≥ 180, irrespective of belief. Therefore, the stage III SE(3) action is for all P2 types to choose y2 = 180, regardless of belief on P1 types. Therefore, the stage II SE(4) action is for all P2 types to accept if x1 ≤ (1000 − (600 − 180)) = 580, regardless of belief on P1 types. Therefore, in MCG(4), at stage I, 31, 41, and 51 (others replaced by Nature) face an 23This uniqueness is only of the SE strategy pro le, not the belief pro le. We mean the same thing by uniqueness in what follows.
45 expected payo of 2×700 3×180 from choosing versus an expected payo 388 (= 5 + 5 ) x1 = 700 of 580 from choosing x1 = 580, given that they must have belief as per the prior distribution in SE(4). Therefore, in SE(4), at stage I, 31, 41, and 51 choose x1 = 580. We note the LFE actions at 4, which contains 0 information set at stage I, 0 information set at stage II, D 31s 22s and 0 information set at stage III, and 0 information set at stage IV. 12s 04s Step 5: x the LFE actions at S4 n, solved above, as Nature's moves to convert n=1 D CG(5) to MCG(5). We now describe the unique SE(5) of MCG(5). All P1 types at stage
V choose x3 = 360, irrespective of belief. Therefore, the stage IV SE(5) action is for all P1 types to accept if y2 ≥ 180, regardless of belief on P2 types. Given this, the stage III SE(5) action is for all P2 types to choose y2 = 180. Therefore, the stage II SE(5) action for all
P2 types is to accept if x1 ≤ 580. Therefore, in SE(5), all P1 types choose x1 = 580 given beliefs determined by the prior distribution on P2 types. We note the LFE actions at D5, which contains 0 information set at stage I, 0 information set at stage II, 0 information 41s 32s 22s set at stage III, 0 information set at stage IV, and 0 information set in stage V. 11s 01s Last step: x the LFE actions at S5 n, solved above, as Nature's moves to convert n=1 D Γ to MCG(6). We now describe the unique SE(6) of MCG(6). In SE(6), all P2 types at stage VI accept x3 ≤ 360, irrespective of belief. Given this, the stage V SE(6) action is for all P1 types to choose x3 = 360, regardless of belief on P2 types. Therefore, the stage IV
SE(6) action is for all P1 types to accept y2 ≥ 360. Note that 32 and 42, at stage III, know that if x1 = 1000, then with probability one, P1's type is 01, who will accept y2 = 180 in stage IV, thus conditional on x1 = 1000, 32 and 42 oer y2 = 180. Conditional on x1 = 700,
32 and 42, at stage III, know that P1's type is 11 or 21 with equal probability. However, the oer of in that case will be rejected with probability 1 , and lead to an expected payo of 180 2 (600−180) 0 , which is less than the payo from oering , and getting a payo of 210 (= 2 + 2 ) 360
240 for sure. Thus, conditional on x1 = 700, 32 and 42 oer y2 = 360. Therefore in SE(6), in stage II, 42 accepts x1 ≤ 760. O equilibrium, 42 believes P1's type must be 51. In stage
I, 51 evaluates the expected payo from x1 = 580, or 700, or 760, given that his beliefs on
46 P2 types are determined by the prior distribution, he chooses x1 = 580 in SE(6). We have now solved for the LFE actions of all the information sets of Γ, which completes the solution stated in Table 1 below.
Table 1: Player-types' LFLE strategies.
Player-type Stage I Stage II Stage III Stage IV Stage V Stage VI
01 x1 = 1000 Y2 = 180 x3 = 360
11 x1 = 700 Y2 = 180 x3 = 360
21 x1 = 700 Y2 = 360 x3 = 360
31 x1 = 580 Y2 = 360 x3 = 360
41 x1 = 580 Y2 = 360 x3 = 360
51 x1 = 580 Y2 = 360 x3 = 360
02 X1 = 700 y2 = 0 X3 = 360
12 X1 = 700 y2 = 180 X3 = 360
22 X1 = 580 y2 = 180 X3 = 360
32 X1 = 580 y2 = 180 if (x1 = 1000), else 360 X3 = 360
42 X1 = 760 y2 = 180 if (x1 = 1000), else 360 X3 = 360
Notes. The table reports the LFE for the Sequential Bargaining game. x1, x3 are the rst and third period demands, respectively, of the rst mover. X1, and X3 are the maximum rst and third period demands, respectively, of the rst mover that are acceptable to the second mover in those periods. y2 is the maximum oer of the second mover to the rst mover from the second period pie. Y2 is the rst mover's minimum acceptable amount from second period pie.
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