Coalitional Stochastic Stability in Games, Networks and Markets

Coalitional stochastic stability in games, networks and markets

Ryoji Sawa∗ Department of Economics, University of Wisconsin-Madison

November 24, 2011

Abstract This paper examines a dynamic process of unilateral and joint deviations of agents and the resulting stochastic evolution of social conventions in a class of interactions that includes normal form games, network formation games, and simple exchange economies. Over time agents unilaterally and jointly revise their strategies based on the improvements that the new strategy proﬁle offers them. In addition to the optimization process, there are persistent random shocks on agents utility that potentially lead to switching to suboptimal strategies. Under a logit speciﬁcation of choice probabilities, we characterize the set of states that will be observed in the long-run as noise vanishes. We apply these results to examples including certain potential games and network formation games, as well as to the formation of free trade agreements across countries.

Keywords: Logit-response dynamics; Coalitions; Stochastic stability; Network formation.

JEL Classiﬁcation Numbers: C72, C73.

∗Address: 1180 Observatory Drive, Madison, WI 53706-1393, United States., telephone: 1-608-262-0200, e-mail: [email protected]. The author is grateful to William Sandholm, Marzena Rostek and Marek Weretka for their advice and suggestions. The author also thanks seminar participants at University of Wisconsin-Madison for their comments and suggestions.

1 1 Introduction

Our economic and social life is often conducted within a group of agents, such as people, firms or countries. For example, firms may form an R & D alliances and found a joint research venture rather than independently conducting R & D. When forming an alliance, two or more firms make a joint decision, or simultaneously make decisions while taking into account other firms’ decisions. In these settings, it is sometimes more appropriate to use a stronger solution concept than Nash equilibrium, one that accounts for joint deviations by groups of agents. In this paper, we introduce an evolutionary equilibrium selection approach for solution concepts stronger than Nash equilibrium, solution concepts that account for coalitional deviations. The importance of stronger solution concepts has been emphasized in recent years due to a grow- ing interest in social and economic networks. This is because Nash equilibrium has relatively weak predictive power in games on networks, particularly network formation games. Examples of such games are co-author networks (e.g. Jackson and Wolinsky (1996)), R & D networks among firms (e.g. Goyal and Moraga-Gonzalez´ (2001)), and trade agreement networks among countries (e.g. Furu- sawa and Konishi (2007)). When link formation between two agents requires both agents’ consent, Nash equilibrium fails to capture the fact that it may be beneficial for two agents to form a link. For instance, it is always a Nash equilibrium for no player to consent to any link, resulting in the empty network. It is thus natural in this context to use a stronger solution concept that is robust against deviations by pairs of agents as in the notion of pairwise stable equilibrium. Once joint deviations are admissible, it is reasonable to consider some settings which are not described by games because some unilateral deviations may be infeasible. For instance, consider a Gale-Shapley marriage problem. Suppose that an individual’s strategy is to choose whom to marry, and that a man and a woman will be married if and only if both agree to do so. This implies that

the set of feasible strategy proﬁles is restricted: letting si denote individual i’s strategy, man m can choose sm = w if and only if woman w chooses sw = m. We call these generalizations of normal-form games interactions. Even when we apply stronger solution concepts to interactions, we may still face the problem of multiple equilibria. For example, in the Gale-Shapley marriage problem, any stable matching is a pairwise stable equilibrium. The average number of stable matchings grows more than proportion- ally as the population increases, making it difﬁcult to predict which will emerge. To address this issue and single out one equilibrium, we apply the concept of stochastic stability. In the stochastic stability approach, we embed a static interaction in a dynamic process in which players revise their strategies based on the improvement that the new strategies offer them relative to the current strategies. We perform a stability test by adding stochastic noise to agents’ strategy choices in the above dynamic and studying the long-run outcomes as noise vanishes. This approach has been well developed to select an equilibrium among the set of Nash equilibria.1 However, despite the importance of decentralized but possibly joint decisions in many social and economic settings, the development of frameworks for analyzing stochastic stability in this setting is in its infancy. A pioneering paper by Jackson and Watts (2002) studied pairwise stochastically stable networks that are immune against random shocks in two players’ decisions. They restricted their

1See Chapter 12 of Sandholm (2010) for an overview.

2 attention to network formation games. The present paper follows a similar line to Jackson and Watts (2002), but extends the range of applications in three directions; (i) applying the stochastic stability approach to a broader class of environments, (ii) extending pairwise stochastic stability to stochastic stability with respect to an arbitrary set of feasible coalitions, and (iii) employing the logit-response dynamic, under which more serious mistakes are less likely to occur. Point (i) allows us to study stochastically stability in a broader class of environments. We define an interaction denoted by I = N, (Si, ui)i∈N , S with player set N, strategy sets Si, payoff functions ui and the set of feasible strategy profiles S ⊆ S = ∏i∈N Si. The restriction on strategy profiles allows us to consider settings where some unilateral deviations are prohibited or infeasible. For example, in an exchange economy, a consumer cannot unilaterally change her allocation, since doing so requires another consumer who is willing to exchange with her. Point (ii) allows us to apply our model in environments where more than two agents may form a coalition. Let R be a family of sets of players, representing the collection of feasible coalitions. We define an R-stable equilibrium of interaction I to be a feasible strategy profile which is immune against deviations by any coalition in R. Suppose for instance that S = S, so that I is a normal form game. Then if R is the family of singletons, R-stable equilibrium corresponds to Nash equilibrium; if R is the family of singletons and pairs, it corresponds to pairwise stable equilibrium; and if R is the family of all nonempty sets of players, it corresponds to strong equilibrium. If not all strategy profiles are feasible, then neither are all unilateral deviations, so some choice of R is necessary to define a meaningful notion of equilibrium. Point (iii) allows us to apply stochastic stability in settings where agents, while sometimes making mistakes, assign smaller probabilities to actions which deliver smaller payoffs. As an example, consider the probability that people send an email to unintended recipients. People will be more careful, and so make such a mistake with lower probability, when their email includes confidential information, e.g. a strategic move of their corporation. Our interest is to characterize strategy profiles that are stochastically stable against not only individual deviations but also coalitional deviations. We introduce a dynamic process in which players randomly form coalitions and revise their strategies in the following way. First, a new strategy profile for them is randomly proposed, and assessed by each agent. In the unperturbed updating process, an agent agrees to the new strategy profile if it yields her higher payoff than the original strategy profile. The new strategy profiles is accepted if all agents in the coalition agree. To this unperturbed dynamic process, we add stochastic noise that leads agents to agree to a new strategy profile according to the logit choice rule. Thus, coalitional deviations sometimes occur even if not all members of the coalition benefit, and this occurs with a probability that declines in the total cost to coalition members. As a consequence, the stochastic process visits every strategy profile repeatedly, and predictions can be made concerning the relative amounts of time that the process spends at each. We consider the behavior of this system as the level of noise in agents’ choice becomes small, defining the stochastically stable strategy profiles to be those which are observed with positive frequency in the long run as noise vanishes. Our paper characterizes the stochastically stable strategy profiles of the logit-response dynamic when coalitional deviations are possible. The intuition for which strategy profiles are stochasti-

3 cally stable comes from the changes in payoffs by switching strategies. Under the logit choice rule, if agents face larger decreases in payoffs by switching to new strategies, it will be less likely that they agree to such revisions. Thus, the decrease in payoffs by a revision represents its unlikeliness. Roughly, the stochastically stable strategy profiles are those that minimize the sum of the payoff decreases by being switched from other strategy profiles. Those strategy profiles are harder to get away from and easier to get back to in the stochastic dynamic. We apply our main result (Theorem 1) to simple exchange economy and show that the stochastically stable allocations are those maximizing the sum of players’ utilities. Note that these allocations are Pareto efficient, but not uniquely so. We generalize our finding by characterizing a class of interactions that exhibit coalitional potential functions, showing that the stochastically stable outcomes are those maximizing the coalitional potential. We use this construction to derive a similar result for the cost-sharing among agents in a fixed network, where every two agents forming a link are required to share the cost of maintaining the link. We show that, in a fixed star network, the greater portion of a link’s cost is sponsored by the peripheral agent as the number of peripheral agents increases. We also apply our main result to a network formation game in which countries form free trade agreements (FTAs). FTA formations have been analysed as network formation games by recent studies, such as Goyal and Joshi (2006) and Furusawa and Konishi (2007). Goyal and Joshi (2006) showed that there are two sets of pairwise stable equilibria; one set represents global free trade, while the other represents divided trading blocs. An unanswered question is whether the decentralized FTA formations will likely lead to global free trade. We answer this question in affirmative; our model shows that a complete network is stochastically stable when the number of countries is a realistic number. Related literature As written above, pairwise stable equilibrium has been often used in the literature on networks. The papers most closely related to this one are Jackson and Wolinsky (1996), Watts (2001) and Jackson and Watts (2002). The notion of pairwise stability is introduced in Jackson and Wolinsky (1996). Watts (2001) and Jackson and Watts (2002) analyze the formation of networks in a dynamic framework. Watts restricted attention to an unperturbed dynamic and discussed whether an efficient network would emerge. Watts found that an efficient network appears less likely as the number of agents increases. Jackson and Watts (2002) adopted the stochastic stability approach of Kandori et al. (1993) and Young (1993) in order to select among pairwise stable equilibria. This approach, known as the best response with mutations, has agents choose every suboptimal strategy with same probability. Although this approach is the most popular one to examine stochastic stability, it does not allow for the possibility that costly mistakes may be less common than minor mistakes. One model that accounts for this possibility is the logit choice rule of Blume (1993) which we adopt here. Under the logit rule, players choose an action according to the logit function, which assigns larger probability to those actions which offer larger payoffs, and so takes into account the magnitudes of payoffs when considering agents’ mistakes. Other closely related papers in the stochastic stability literature are Alos-Ferrer´ and Netzer (2010), Kandori et al. (2008) and Newton (2011). Alos-Ferrer´ and Netzer (2010) characterize the stochastically stable outcomes of the logit response dynamic in finite normal form games. Like our model,

4 their model handles deviations by more than one agent; but unlike in our model, these agents do not account for the new choices of simultaneously revising agents. Thus, Alos-Ferrer´ and Netzer (2010) examined stochastic stability of Nash equilibrium by introducing simultaneous unilateral deviations, while my model examines stochastic stability of R-stable equilibrium by introducing joint deviations. Kandori et al. (2008) employed a similar logit dynamic to study a simple exchange economy showing that stochastically stable allocations are those maximizing the sum of players’ utility functions. As written above, we deﬁne a class of interactions that exhibit coalitional potentials, and show that the economies considered in Kandori et al. (2008) fall into this category. Newton (2011) modiﬁes the stochastic stability approach of Kandori et al. (1993) and Young (1993) by introducing vanishingly small probabilities of coalitional deviations, and uses this approach to examine the robustness of Nash equilibrium. In contrast, we assume that forming a coalition is a likely event and focus on stochastic stability of stronger solution concepts than Nash equilibrium. The paper is organized as follows. Section 2 describes the model. Section 3 introduces the dynamic process and offers the characterization of stochastically stable states. Applications of our framework are laid out in Section 4. Section 5 concludes.

2 The Model

2.1 Interactions Let G = N, (Si, ui)i∈N denote a normal-form game with player set N = {1, . . . , n}, finite strategy sets Si and payoff functions ui. Let S = ∏i∈N Si denote the set of pure strategy profiles. The payoff functions are defined as ui : S → R. We define an interaction to be a collection I = N, (Si, ui)i∈N , S where S ⊆ S is the set of feasible strategy profiles for the interaction. In interaction I, the payoff functions need only be defined on

S , i.e. ui : S → R. Since S may be a strict subset of S, interactions include not only normal-form games, but also more general multi-agent choice environments.

For a given strategy si ∈ Si of player i, let S−i(si) = {s−i ∈ ∏j6=i Sj : (si, s−i) ∈ S } denote the set of pure strategy profiles of i’s opponents that are feasible in combination with si. For a coalition J ⊆ N and strategy profile s ∈ S , write sJ and s−J to represent a strategy profile of agents in J and J not in J respectively. Let S (s−J) = {sJ ∈ ∏i∈J Si : (sJ, s−J) ∈ S } denote the set of pure strategy −J profiles of players in J that are feasible in combination with s−J. Similarly, let S (sJ) = {s−J ∈

∏i∈/J Si : (sJ, s−J) ∈ S }.

2.2 Feasible Coalitions and R-Stable Equilibria

We allow agents to jointly revise their strategies by forming a coalition. Which coalitions are feasible may vary among applications. For example, it would be appropriate that a set of feasible coalitions is a set of single agents and pairs in network formation games, as in pairwise stable equilibrium. However, that set may not be appropriate in some exchange economies in which more than two agents may form a coalition.2 To meet the variety of needs across applications, we introduce a

2For example, there has been three-way exchanges in kidney exchange, i.e. three patients exchange their kidney-donors. See Roth et al. (2007).

5 ﬂexible solution concept. Let R ⊆ P (N) \{φ} denote a class of feasible coalitions, where P (N) denotes the power set of N and φ denotes empty set. The following solution concept describes a strategy proﬁle such that for any coalitional deviation by J ∈ R, there is at least one agent in J who will not be better off. In the strict formation, there is at least one agent in J who will be worse off.

0 Deﬁnition 1. A strategy proﬁle s ∈ S is an R-stable equilibrium in (I, R) if for all J ∈ R and all sJ ∈ J S (s−J), 0 ui (s) ≥ ui sJ, s−J for some i ∈ J.

0 J A strategy proﬁle s ∈ S is a strict R-stable equilibrium in (I, R) if for all J ∈ R and all sJ ∈ S (s−J),

0 ui (s) > ui sJ, s−J for some i ∈ J.

Note that R-stable equilibrium is equivalent to Nash equilibrium when R = N is the set of single agents. R-stable equilibrium is equivalent to a version of pairwise stable equilibrium when R = {J ⊆ N : |J| ≤ 2}.3 R-stable equilibrium is equivalent to strong equilibrium (Aumann (1959)) when R = P (N) \{φ}.

2.3 Examples of Interactions

To illustrate the sorts of environments represented by interactions, we introduce three examples. Example 1 (Normal-form games). When S = S, interaction I = N, (Si, ui)i∈N , S corresponds to normal-form game G = N, (Si, ui)i∈N . Note that when R = N in Example 1, (I, R) corresponds to a noncooperative normal form game. The next example shows a network formation game where both agents need to consent to form a link.

Example 2 (Network formation games). A strategy of agent i in interaction I is her choice of a vector n−1 of contributions si ∈ Si = {0, 1} . Let sij be an entry of si and representing agent i’s contribution to forming link ij. Link ij is formed if and only if sij = sji = 1. Payoffs derive from two sources: the collaborative work among connected agents and the costs of maintaining links. The feasible strategy proﬁle space is given by S = S. A network g is a set of links formed by agents. Let s ∈ S be a strategy proﬁle and g (s) the network generated by s, that is, (ij) ∈ g(s) if and only if sij = sji = 1. The payoffs of agent i are given by

ui (si, s−i) = ϕ (g (s) , si) (1) where ϕ is a function that maps a network and the agent’s strategy to her payoffs.

In Example 2, it is natural to imagine that a pair of agents will form a link if it is beneﬁcial for both agents. Then, a reasonable solution concept is R-stable equilibrium with R = {J ⊆ N : |J| ≤ 2}. The next example shows an interaction which is not a normal form game, i.e. S ⊂ S.

3A pairwise stable equilibrium requires that there is no pair of players such that one player be strictly better off and the other be weakly better off. For variations, see Section 5 of Jackson and Wolinsky (1996).

6 Example 3 (Simple exchange economies). We deﬁne an interaction I to represent a housing market as in Shapley and Scarf (1974). Let H denote the ﬁnite set of (indivisible) houses with |H| = n. Let the strategy space for agent i be ( )

n h Si = si ∈ {0, 1} ∑ si = 1 h∈H

h Strategy si can be interpreted as a house allocation for agent i. An entry si = 1 in si ∈ Si represents agent i owning house h. Deﬁne the set of feasible strategy proﬁle as ( )

h S = s ∈ S ∑ si = 1 ∀h ∈ H . i∈N

In words, S is the set of allocations in which each owner has exactly one house. Note that under S , no single agent can unilaterally change her strategy or her allocation. Thus, to study behavior in this interaction, we need to specify which groups of owners may jointly change their strategies by swapping their houses.

3 Dynamics and Stochastically Stable States

In this section, we apply the stochastic stability approach to interaction I with families of feasible coalitions. This is brieﬂy summarized as follows. In this approach, we embed a static interaction in a dynamic process in which agents randomly form coalitions and revise their strategies based on improvements in their payoffs in the presence of stochastic payoff shocks. We examine the limiting probability distribution over strategy proﬁles as the level of stochastic shocks approaches zero.

3.1 The Unperturbed Dynamic

We describe an unperturbed dynamic, i.e. a dynamic with no stochastic payoff shocks. The state of the system st in period t is a vector of the strategies played by each agent and is deﬁned by

t t t s = s1,..., sn ∈ S

t where si denote the strategy played by agent i in period t. The dynamic interaction proceeds as follows. In each period t, a set of agents, denoted by J, is randomly selected from R to revise their 0 J t strategy. For agents in coalition J, one feasible strategy profile sJ ∈ S (s−J) is proposed at random. 0 An agent will agree to sJ if she will be strictly better off by switching. If an agent is indifferent between the current strategy and the proposed strategy, she will agree with probability α ∈ (0, 1). 0 0 Otherwise, she disagrees. If all agents in J agree with playing sJ, agents will switch to sJ for the next round. If at least one agent disagrees, then no agent will alter his current strategy. The dynamic above determines a Markov chain with state space S . Note that we use state s ∈ S and strategy profile s ∈ S interchangeably since these two sets coincide. Now, we discuss the 0 transition probabilities of the Markov chain for two arbitrary strategy profiles s, s ∈ S . Let qJ

7 denote the probability that exactly players in coalition J receive revision opportunities.4 We assume

that qJ > 0 for all J ∈ R. In words, coalition J has positive probability to revise in the dynamic if J 0 is feasible in interaction I. qs0 (J, s) denotes the probability that s is chosen as a new strategy profile when agents form coalition J in state s. We sometimes call R the family of revising sets in the sense that each element of R is the set of agents simultaneously revising their strategy profile. For any two 0 0 strategy profiles s, s ∈ S , let Rs,s0 = J ∈ R sk = sk ∀k ∈/ J denote the set of coalitions potentially leading from s to s0. The transition probabilities for (s, s0) of the unperturbed dynamic is given by " # 0 0 Ps,s0 = ∑ qJ qs0 (J, s) ∏Ψ uj (s) , uj sJ, s−J , (2) J∈Rs,s0 j∈J

where  0 x > y  Ψ (x, y) = α x = y   1 x < y.

0 We say that state s is absorbing in the unperturbed dynamic if Ps,s = 1, i.e. the process does not exit s. We also say that a nonempty set of states RC is a recurrent class in the unperturbed dynamic if there is zero probability that the unperturbed dynamic can cause the process to exit RC and a positive probability of moving from any state in RC to any other state in RC in a ﬁnite number of time periods. Note that a singleton recurrent class must consist of an absorbing state. The following proposition relates an absorbing state with a strict R-stable equilbrium.

Proposition 1. State s is absorbing in the unperturbed dynamic if and only if s is a strict R-stable equilbrium in (I, R).

Proof. The ’if’ part, recall that if s is a strict R-stable equilbrium, then there exist at least one agent who will be strictly worse off for any coalitional deviation. Then, equation (2) implies that the transition probabilities from s to any other states are zero. For the ’only if’ part, suppose not. There exists absorbing state s that is not a strict R-stable equilbrium. Then, there exists J ∈ R such that all agents in J will be weakly better off by some joint deviations. Equation (2) implies that state s is not absorbing, a contradiction.

The above proposition tells us that the unperturbed process starting from a state that is a strict R-stable equilibrium does not exit the state. Also note that the unperturbed dynamic starting from any state must lead to a strict R-stable equilibrium or a recurrent class consisting of multiple states (where these states are repeatedly visited). This is because the unperturbed process cannot exit a recurrent class if it starts there. This suggests that the unperturbed dynamic does not provide unique predictions. The reason is that it does not distinguish any strict R-stable equilibrium even if these equilibria may differ in the robustness against some exogenous events, e.g. errors made by agents. The approach introduced below can be thought of a robustness check for equilibria. In what follows, we add stochastic shocks to the unperturbed dynamic and examine how resilient these equilibria and cycles are to stochastic shocks.

4 Probability qJ is independent of time, but it may depend on the current state. All the results will hold without change if qJ > 0 for one state implies that qJ > 0 for any state.

8 3.2 The Coalitional Logit Dynamic

We now formally describe a stochastic dynamic behavior of interaction I. We focus on a logit- response dynamic of Blume (1993). As an example of the logit rule, suppose that the current strategy 0 profile is given by s and the set revising agents is given by J. Also suppose that sJ is proposed as 0 their new strategy profile. The probability that all agents in coalition J agree with strategy profile sJ is given by

h −1 0 i exp η ui sJ, s−J 0 Pr members in J agree with sJ s = h i . (3) ∏ −1 0 −1 i∈J exp η ui sJ, s−J + exp [η ui (s)]

where parameter η ∈ (0, ∞) denotes the noise level of the logit-response function and we assume that η is small. Note that agent i takes into account other agents’ new strategies in the LHS of equation (3). The logit response dynamic is a Markov chain on the state space S with stationary transition probabilities. The probability for transition (s, s0) is given by h i exp η−1 u s0 , s η ∑j∈J j J −J P = q q 0 (J, s) . (4) s,s0 ∑ J s h h i i ∈ −1 0 −1 J Rs,s0 ∏j∈J exp η uj sJ, s−J + exp η uj (s)

Note that the unperturbed dynamic is given by the limiting dynamic as η approaches zero.5

3.3 Limiting Stationary Distributions and Stochastic Stability

Let πη denote the stationary distribution of the Markov chain with noise level η. Also let πη(s) denote the probability that πη places on state s. πη(s) represents the fraction of time in which state s is observed over a long time horizon. It is also the probability that state s will be observed at any given time t, provided that t is sufﬁciently large. Thus, the agents’ behavior is nicely summarized by πη in the long-run. We say that state s is R-stochastically stable if the limiting stationary distribution places positive probability on s.

η Definition 2. State s is R-stochastically stable if limη→0 π (s) > 0. To determine which states will be observed often in the long run, we now introduce several definitions in order to compute the unlikeliness of transitions. Given a state s, define a s-tree to be a directed graph T such that there exists a unique path from any state s0 ∈ S to s. The following definition is due to Alos-Ferrer´ and Netzer (2010). 0 Definition 3. A revision s-tree is a pair (T, ρ) where (i) T is an s-tree, (ii) (s, s ) ∈ T only if Rs,s0 6= φ, and 0 0 (iii) ρ : T → R is such that ρ (s, s ) ∈ Rs,s0 for all (s, s ) ∈ T. Let T (s) denote the set of revision s-trees. The waste of a revision tree (T, ρ) ∈ T (s) is defined as n 0 o W (T, ρ) = ∑ ∑ max uj (s) − uj sJ, s−J , 0 . (5) (s,s0)∈T j∈J

5More precisely, the unperturbed dynamic with α = 1/2 is given by the limiting dynamic as η approaches zero. However, observe that the set of recurrent classes in the unperturbed dynamic does not differ for α ∈ (0, 1). Our analysis will not differ for all α ∈ (0, 1).

9 where J = ρ (s, s0) for (s, s0) ∈ T. Note that the main difference in the waste from Alos-Ferrer´ and 0 Netzer (2010) is that each agent j ∈ J assesses a coalitional deviation sJ instead of an individual deviation. Roughly, the waste is the sum of the decreases in agents’ payoffs along the tree. A stochastic potential of s-tree is deﬁned as W (s) = min W (T, ρ) . (T,ρ)∈T (s) We cannot directly apply the stochastic potential function deﬁned in Alos-Ferrer´ and Netzer (2010), although the analysis is carried out in a similar way proposed in their paper. Agents does not consider the effect of strategy simultaneously chosen by other agents in Alos-Ferrer´ and Net- zer (2010). By contrast, revising agents take into account their revising actions in this paper. As η approaches zero, the stationary distribution converges to a unique limiting stationary distribution. Our main result is the following theorem which offers the characterization of R-stochastically stable states.

Theorem 1. A state is R-stochastically stable if and only if it minimizes W (s) among all states.

Its proof and most of profs in this paper are relegated to Appendix.

3.4 Conditions for Selection among R-stable Equilibria

Predictions of the stochastic stability approach are made concerning the relative amounts of time that the stochastic dynamic process spends in each state. Since the stochastic dynamic process gravi- tates to recurrent classes of the unperturbed dynamic, the limiting stationary distribution may place positive probability on non-singleton recurrent classes. Since a recurrent class does not necessarily correspond to an R-stable equilibrium, the dynamic of our model may not always select R-stable equilibrium In this section, we characterize the class of games in which our model selects one or some R-stable equilibria given the set of feasible coalitions R. Our model selects a subset of R-stable equilibria if there does not exist any non-singleton recurrent class. We characterize a sufficient condition in terms of payoffs under which a subset of R-stable equilibria is selected. This condition corresponds to the acyclic condition of Young (1993) when S = S and R = N. We define an weakly improving path as follows, and define a cycle of better responses.

Definition 4. An weakly improving path from a strategy profile s1 to another strategy profile sK is a sequence of strategy profiles {s1, s2,..., sK}, such that, for all k = 1, . . . , K − 1,

(i) Rsk,sk+1 is not empty, and

k+1 k (ii) ∃J ∈ Rsk,sk+1 such that ui(s ) ≥ ui(s ) ∀i ∈ J.

An weakly improving path {s1,..., sK} is a sequence of strategy profiles that will be observed with positive probability in an unperturbed dynamic starting from s1. A set of strategy profiles C with |C| ≥ 2 form a cycle if there exists an weakly improving path connecting a strategy profile s to another strategy profile s0 for all s ∈ C and s0 ∈ C. A cycle C is closed if there is no weakly improving path from any s ∈ C that leads to some s0 ∈ C. We call (I, R) R-acyclic if (I, R) has no closed cycle.

10 It is clear that a non-singleton recurrent class corresponds to a closed cycle. If the interaction is R- acyclic, then the process has only singleton recurrent classes (strict R-stable equilibria), and selects a subset of them. It is formally shown by the following proposition.

Proposition 2. Suppose that (I, R) is R-acyclic. Then, any stochastically stable state is a strict R-stable equilibrium.

3.5 A Radius-Coradius theorem

Therem 1 provides an algorithm to determine R-stochastically stable states. However, it might be a burden to compute a stochastic potential of each state. In the stochastic stability literature, several methods have been developed to reduce these computational burdens. One of the most powerful methods to determine stochastically stable states is the Radius-Coradius theorem of Ellison (2000). In this section, we prove an analogous result to the Radius-Coradius theorem in Ellison (2000). In Section 4.4, we show that our result is useful to determine R-stochastically stable states.

A directed graph d (s1, sk) on S is a path if d (s1, sk) is a finite, repetition-free sequence of transitions {(s1, s2) , (s2, s3) ,..., (sk−1, sk)} such that si ∈ S for all i = 1, . . . , k. A path d (s1, sk) is feasible if Rsisi+1 is not empty for all i = 1, . . . , k − 1. Define a revision path as a pair (d (s1, sk) , ρ) such that 0 ρ (si, si+1) ∈ Rsi,si+1 for all (si, si+1) ∈ d (s1, sk). Let D (s, s ) be the set of all revision paths with initial point s and terminal point s0. Let the waste W (d(s, s0), ρ) be the waste of (d(s, s0), ρ) as a revision tree. The basin of attraction of a state s, B (s) ⊆ S , is the set of all states s0 such that there exists a path (d, ρ) ∈ D (s0, s) with W (d, ρ) = 0. The radius of state s is defined as

0 0 R (s) = min W (d, ρ) s ∈/ B (s) , (d, ρ) ∈ D s, s . (6)

Also the coradius of state s is deﬁned as

00 0 00 CR (s) = max min W (d, ρ) s ∈ B (s) , (d, ρ) ∈ D s , s . (7) s0∈/B(s)

Let U denote an arbitrary set of states. Similarly to B (s), B (U) ⊆ S is the set of all states s0 such that there exists a path (d, ρ) ∈ D (s0, s) with W (d, ρ) = 0 for some s ∈ U. The radius of U and the coradius of U are deﬁned as

0 0 R (U) = min min W (d, ρ) s ∈/ B (U) , (d, ρ) ∈ D s, s , (8) s∈U 00 0 00 CR (U) = max min W (d, ρ) s ∈ B (U) , (d, ρ) ∈ D s , s . (9) s0∈/B(U)

R (U) is the minimum waste for the state to move away from the basin of U. CR (U) is the maximum waste for the state to move into the basin of U.

Theorem 2 (Radius-Coradius). Let U denote the set of absorbing states of the unperturbed dynamic. Sup- pose a subset of absorbing states U1 ⊂ U such that R (U1) > CR (U1). Then, the limiting stationary distribution places probability one on U1.

11 In the remainder of this paper, we study applications of our model.

4 Applications

4.1 Interactions with Coalitional Potentials

In this section, we introduce a class of interactions which exhibit a function named coalitional potential. Then, we show that the stochastically stable strategy profiles are those maximizing coalitional potential in this class of interactions. This result provides a much simpler way to determine the stochastically stable strategy profiles relative to Theorem 1. Let I = N, (Si, ui)i∈N , S be an interaction and R the set of feasible coalitions. We define a coalitional potential and a coalitional potential interaction as follows.

0 J Deﬁnition 5. A function P : S → R is a coalitional potential for (I, R) if for all J ∈ R, sJ, sJ ∈ S , −J s−J ∈ S 0 0 ∑ uj sJ, s−J − uj sJ, s−J = P sJ, s−J − P sJ, s−J . j∈J Deﬁnition 6. (I, R) is called a potential interaction if it admits a coalitional potential.

Note that a coalitional potential function is a potential function in the sense of Monderer and Shapley (1996) when S = S and R = N. In the following sections 4.2 and 4.3, we show that this class of coalitional potential interactions includes interesting examples. The following lemma relates the maximizer of P with R-stable equilibrium of I. And the subsequent proposition shows that our model always selects R-stable equilibrium that maximizes P.

Lemma 1. Suppose that (I, R) is a coalitional potential interaction with potential P. A strategy proﬁle s∗ ∈ S is an R-stable equilibrium if s∗ maximizes P.

J Proof. For any J ∈ R and sJ ∈ S , observe that

∗ ∗ ∗ ∗ ∑ ui sJ, s−J − ui (s ) = P sJ, s−J − P (s ) ≤ 0. i∈J

∗ ∗ This implies that if there exists agent i ∈ J such that ui sJ, s−J > ui (s ), then there must exist at ∗ ∗ ∗ least one agent j ∈ J such that uj sJ, s−J < uj (s ). According to Deﬁnition 1, s is an R-stable equilibrium

Note that the converse of the lemma is not necessarily true. There may exist an R-stable equilibrium that does not maximize P. A similar fact is well known for potential games. For coalitional potentials, see the example below.

Example 4. Suppose that interaction I with N = {1, 2}, Si = {A, B} and S = {(A, A), (B, B)} and that R = {(1, 2)}. Utility functions are given by u1(A, A) = 5, u1(B, B) = 3, u2(A, A) = 1, and u2(B, B) = 4. The interpretation of this interaction is that two people choose which restaurant A or B to go together. They have different preferences over restaurants. Observe that the coalitional potential of (I, R) is given by u1 + u2. Both strategy proﬁles are R-stable equilibria, but (A, A) does not maximize the potential.

12 The following proposition, together with the lemma above, shows that our model always selects a particular subset of R-stable equilibria.

Proposition 3. Suppose that (I, R) is a coalitional potential interaction with potential P. State s∗ is R- stochastically stable if and only if s∗ maximizes P.

Proposition 3 is a generalization of results of Blume (1997) which considered potential games, i.e. S = S and R = N.

4.2 An application in simple exchange economy

In this section, we consider an simple exchange economy of Kandori et al. (2008). We use this economy as an example of an interaction that exhibits a coalitional potential. In an exchange economy, coalitions are naturally formed, since two or more agents are involved with an exchange of goods. Following Kandori et al. (2008), we suppose a house swapping market introduced by Shap- ley and Scarf (1974). Let H denote the ﬁnite set of houses with |H| = n. Let the strategy space for agent i be ( )

n h Si = si ∈ {0, 1} ∑ si = 1 h∈H

h An entry si = 1 in si ∈ Si represents that agent i owns house h. Strategy si can be interpreted as a house allocation for agent i. Let ( )

h S = s ∈ S ∑ si = 1 ∀h ∈ H . i∈N

In words, S is the set of states in which every agent has exactly one house. Thus, strategy proﬁle space S can be interpreted as the space of feasible allocations. The utility of agent i is given by ( ) = h = ∈ R ui s ui h si 1 for s S . Let revising set be such that the perturbed dynamic visits every allocation, or state, with positive probability. The dynamic interaction proceeds as follows. At the beginning of each period, a coalition J ∈ R is chosen. They choose one possible allocation 0 J sJ ∈ S s−J where s denotes the current state, or the current allocation, and

J 0 S s−J = s ∈ S : si = si ∀i ∈/ J . (10)

0 0 The probability that s is chosen is exogenously given by qs0 (s, J) such that qs0 (s, J) = 0 if s ∈/ J 0 S s−J . Agents in J agree with s with the probability given by (4). Otherwise, the trade does not occur in a given period. We have the following characterization of R-stochastically stable allocations.

Proposition 4. The limiting stationary distribution places probability one on the set of allocations that maximize the sum of the agents’ utility functions.

A version of the result has been proved by Kandori et al. (2008). However, the above proposition provides slightly more general result in the sense that it does not require the following restrictions of Kandori et al. (2008); (i) probability qJ must be independent of the current strategy proﬁle, and

(ii) probability qs0 (J, s) must be symmetric in the candidate strategy distribution, that is, qs0 (J, s) = 0 qs(J, s ) must hold.

13 Restriction (i) rules out settings where coalition formations depend on the strategy profile. For instance, suppose a listing service, e.g. claigslist.org, that provides a list of potential traders. When people use a listing service to find their counter-part, the coalition formations may depend on the current allocation. This is because a typical listing service provides a list sorted by categories, e.g. houses sorted by location. It implies that the probability people find their counter-part on such listing services would depend on the current allocation. One implication of restriction (ii) is that there cannot exist any favorable choice of strategy profiles. To see this, observe that

0 ∑ qs0 (J, s) = 1 ⇔ ∑ qs(J, s ) = 1. 0 J 0 J s ∈S (s−J ) s ∈S (s−J )

J 1 2 3 1 2 3 For the intuition, suppose J = 1, 2, 3, S (s−J) = {s , s , s }, and that s Pareto domitnes s and s . Then, it is natural to imagine that people pick up s1 with higher probability when the allocation 2 3 2 3 is s or s , i.e. qs1 (J, s) > 0.5 for s ∈ {s , s }. However, restriction (ii) rules out these probability distributions.

4.2.1 Assigning a utility function to preference order

Ben-Shoham et al. (2004) considered a house allocation problem as a triple N, H, {i}i∈N where i is a complete, transitive preference order over H. Let hi (s) denote the house allocated for agent i in allocation s. In their model, the probability that agent i and j trade is given by εni+nj where

nx = h ∈ H : hx (s) x h x hy (s) for (x, y) ∈ {(i, j) , (j, i)} . (11)

The envy level of agent i in allocation s is deﬁned as ei (s) = |{h ∈ H : h x hx (s)}|, that is the number of houses agent i prefers to his current house. Ben-Shoham et al. (2004) showed that the minimum ni+nj envy allocation is selected when the transition probability is given by ε with nx deﬁned in (11). The present model can yield the exactly same result by assigning the following utility to each house allocation:

ui (s) = |{h ∈ H : hi (s) x h}| .

It is obvious that ui (s) correctly represents the agent i’s preference. Following Ben-Shoham et al. (2004), let R = {J ⊂ N : |J| = 2}. Proposition 4 tells us that states maximizing the sum of utilities are stochastically stable. Observe that ( ) ( ) ∗ ∗ 0 ∗ ∗ 0 s : ui (s ) = max ui s = s : [H − ui (s )] = min H − ui s ∑ 0∈ ∑ ∑ 0∈ ∑ i∈N s S i∈N i∈N s S i∈N ( ) ∗ ∗ 0 = s : ei (s ) = min ei s . ∑ 0∈ ∑ i∈N s S i∈N

Thus, our model provides the same prediction as Ben-Shoham et al. (2004) by assigning a linear utility to the preference order.

14 4.2.2 Exchange economy with K goods

It is straightforward to extend the analysis to an exchange economy with more than one good. Suppose an economy consisting of consumer set N = {1, . . . , n} and K goods. Consumer i’s con- 1 K K 6 k sumption bundle is denoted by si = {si ,..., si } ∈ Si = Z+. An entry si ≥ 0 represents agent i’s holdings of good k. The total amount of each good k, or the total endowment of k, is denoted by ek ≥ 0 for k = 1, . . . , K. Let the space of feasible allocations S be ( )

k S = s ∈ S ∑ si = ek ∀k = 1, . . . , K . i∈N

We assume that the intrinsic utility of agent i is given by ui (s) = ui (si). Again, let the class of feasible coalitions R be such that the perturbed dynamic visits every allocation with positive probability. The dynamic interaction proceeds similarly. At the beginning of each period, a coalition J ∈ R is chosen. 0 J J Consumers in J choose one allocation sJ ∈ S s−J where S s−J is given by (10). They will agree to the exchange and switch to allocation s0 if every consumer yields higher utility subject to the random shocks. Deﬁne the social surplus of this economy as SS(s) = ∑i∈N ui(s). According to a similar discussion to the house-exchange economy, it is obvious that SS(s) is a coalitional potential of this interaction. Then, the following proposition characterizes R-stochastically stable allocations. Since the proof is straightforward, it is stated without proof.

Proposition 5. Suppose an exchange economy consisting of n consumers and K goods. The limiting stationary distribution places probability one on the set of allocations that maximize the social surplus.

The proposition holds even if utility function is decreasing over some goods. However, note that the endowment constraints must hold for all goods. An interpretation of such a good is that it might be a ’task’ that must be carried out by somebody. The following example illustrates the case.

Example 5 (Divide the household chores). Two individuals 1 and 2 share their monthly (30 days) household duties, say cooking and dish washing denoted by c and d respectively. Assume that each x x duty must be done everyday so that ∑i=1,2 si = 30 holds for x ∈ {c, d}. Also assume that si ∈ Z for i ∈ {1, 2} and x ∈ {c, d}. Suppose their utility functions are given by

c d u1 = α1 log (30 − s1) + (1 − α1) log (30 − s1) c d u2 = α2 log (30 − s2) + (1 − α2) log (30 − s2) where 0 < αi < 1 for i ∈ {1, 2}. αi is interpreted as the relative dislike of cooking to dish washing for individual i. At the beginning of each month, they negotiate to swap their duties. Assuming that

α1 = 2/3 and α2 = 1/3, the R-stochastic stable allocation is given by

c 30α2 d 30α1 s1 = = 10, s1 = = 20. α1 + α2 α1 + α2

6Each good is divisible (but not perfectly).

15 4.3 Cost sharing in a ﬁxed network: distance-based payoffs

In this section, we study cost-sharing among agents in a ﬁxed network, denoted by g, with the distance-based payoffs. Network g is a set of links and we write (ij) ∈ g if there is a link between

agents i and j in g. The set of neighbors of agent i denoted by gi is the set of agent with which agent i has a link. We say that i1 and ik are connected if there exists a set of distinct agents {i1, i2,..., ik} such that {i1i2,..., ik−1ik} ⊂ g. Let sij denote agent i’s contribution to link (ij) ∈ g. For all (ij) ∈ g, a strategy space of agent i with regards to link (ij) is such that Sij = {0, γ, 2γ,..., c − γ, c} where γ > 0. Note that no element in Sij is = × negative, i.e. there is no transfer between agents. Agent i’s strategy space is defined as Si j∈gi Sij. Since links are fixed, a strategy profile s = (s1,..., sn) is such that sij + sji = c > 0 for any (ij) ∈ g. Since a change in cost-sharing must involve two agents, we define R = {J ⊂ N : |J| = 2}. Let S denote the set of feasible strategy profiles, that is

S = s ∈ ×i∈NSi : sij + sji = c ∀ (ij) ∈ g .

Let d (i, j) denote the distance between i and j in terms of the number of links in the shortest path between them. Let d (i, j) = ∞ if i and j are not connected. A utility function is distance-based if ! d(i,j) ui (si, s−i) = ϕ ∑ δ − ∑ sij j6=i j∈gi

for some δ < 1 and an increasing function ϕ : R → R. The interpretation of the utility function is that agent i joined some collaborative work if she has an access to or is being accessed by another agent. The following lemma is immediate, thus provided without proof.

Lemma 2. Any state s is a singleton absorbing state of the unperturbed dynamic.

The lemma implies that there is no cycle in the unperturbed dynamic.

Proposition 6. A state s is R-stochastically stable if and only if s maximizes the sum of the agents’ utilities.

Proof. Note that a change in strategies sij and sji affects only agents i and j’s utility functions. Then, the cost sharing interaction admits a coalitional potential P such that for all s ∈ S

P (s) = ∑ ui (s) . i∈N

The claim follows from Proposition 3.

The characterization of stochastically stable states depends on the form of utility function. The following corollary is immediate.

Corollary 1. If ϕ is linear, all states are R-stochastically stable. If ϕ is concave, any R-stochastically stable state s satisﬁes the following properties: for all (ij) ∈ g

Yi (s) − Yj (s) ≤ γ if sij > 0 and sji > 0,

Yj (s) − Yi (s) < γ if sij = c and sji = 0,

16 where d(i,j) Yi (s) = ∑ δ − ∑ sij. j6=i j∈gi

Proof. It is obvious if ϕ is linear. For the concave case, suppose s ∈ S such that Yi (s) − Yj (s) > γ 0 0 and sij, sji > 0 for some i, j. Let sij = sij + γ and sji = sji − γ. The concavity implies that

ϕ (Yi (s)) − ϕ (Yi (s) − γ) < ϕ Yj (s) + γ − ϕ Yj (s) 0 0 ⇔ ui (s) − ui sij, s−ij < uj sji, s−ij − uj (s) .

Thus, s does not maximize P. We can prove the second inequality similarly.

4.3.1 Cost-sharing in a star network

A network is called a star network if one agent (the center agent) has links with every other agent and every other agent has no link except the one with the center agent. Many studies in the literature in network formations have shown that a star network is the most likely structure of networks when payoffs are distance-based.7 These suggest that we can expect that agents will eventually form a star network. We consider the cost-sharing among agents after agents have settled in a star network. Suppose network g is a star network and that the agents’ payoff functions are distance-based and concave. We assume that δ − δ2 < c.8 Using Proposition 6, we have the following corollary.

Corollary 2. In a ﬁxed star network, the center agent will contribute either k∗γ or (k∗ + 1) γ to each link in R-stochastically stable states where k∗ is given by an integer satisfying

(n − 2) δ − δ2 + c (n − 2) δ − δ2 + c − 1 < k∗ ≤ . γn γn

It might be of interest that k∗ approaches zero as δ approaches one with sufﬁciently large n. If the good or information does not depreciate through links, e.g. digital data, then the star network would be mainly supported by peripheral agents.

4.3.2 Extension to positive transfers and egalitarian outcomes

Let Sij = {γz : z ∈ Z} for some γ > 0 and all (ij) ∈ g. Similarly to Corollary 1, we have the following corollary. We omit its proof since it is straightforward.

Corollary 3. If ϕ is linear, all states are R-stochastically stable. If ϕ is concave, R-stochastically stable state s satisﬁes the following property: for all (ij) ∈ g

Yi (s) − Yj (s) ≤ γ.

The corollary implies that the utility levels of any connected agents becomes closer if γ is smaller.

7See Bala and Goyal (2000), Feri (2007) and Hojman and Szeidl (2008), for example. 8It is a known condition under which a star network will be stochastically stable (see Feri (2007)).

17 4.4 An application in network formation of trade agreements

In this section, we examine an application of the network formation model, namely network formation of bilateral trade agreements. The model is due to Goyal and Joshi (2006). There is a set of countries, denoted by N = {1, . . . , n}, each of which has one firm. We call the single firm in country i firm i. Each firm can sell in the domestic market as well as foreign markets. A non-directed network g is a set of undirected links of pairs of countries. A link between countries i and j, denoted by (ij) ∈ g, means that a free-trade agreement (or FTA) is established between i and j. We say that firm

i has a tariff-free access to country j and write gij = 1 when (ij) ∈ g. The set of neighbors of country i denoted by gi is the set of countries to which firm i has a tariff-free access. Let ni = |gi| + 1 be the number of countries (including country i) that firm i has a tariff-free access. Network formation interaction I is defined as follows. A strategy of agent i in I is her choice of a n−1 vector of contributions si ∈ Si = {0, 1} . Let sij for j ∈ N \{i} be an entry of si and represents the agent i’s contribution to forming a link ij. Link ij is formed if and only if sij + sji = 2. The payoffs, or social welfare, of each country are defined in the next section. The strategy profile space is defined as

S = s ∈ ×i∈NSi sij = sji ∈ {0, 1} .

Note that S implies that i ∈ nj if and only if j ∈ ni. We consider bilateral free trade agreements and let R = {J ⊂ N : |J| ≤ 2} be the class of feasible coalitions. Also, for any s, s0 ∈ S , let n o 0 0 0 Rs,s0 = J ∈ R si − si ≤ 1 if i ∈ J, sij = sij if i, j ∈/ J, and sij ≤ sij if i ∈ J, j ∈/ J .

This revising set implies that at most one FTA may be established or terminated in each period. Furthermore, it implies that both countries must agree with establishing an FTA, while one country can unilaterally terminate its FTA. Let g (s) the network emerged from s ∈ S . Observing that there is a one-to-one mapping between s ∈ S and g (s), we sometimes write g to represent state s such that g = g (s).

4.4.1 Supply, Demand and Social Welfare

In each country there is a single firm producing a homogeneous good and competing as a Cournot j oligopolist in all countries. Let Qi denote the output of firm j in country i. The total supply in country i is given by j Qi = ∑ Qi. j∈N In each country i ∈ N, a firm faces an identical inverse linear demand given by

Pi = α − Qi for α > 0.

All ﬁrms have a constant and identical marginal cost of production γ > 0. We assume that α > γ. Following Goyal and Joshi (2006), we assume that a ﬁrm i sells in country j if and only if there is

18 9 i a FTA. In Cournot competition, the quantity sold by firm i is given by Qj = (α − γ) / nj + 1 if (ij) ∈ g. The social surplus of country i given network g, denoted by SSi(g), is the sum of consumer surplus and firm’s profits. It is written as:

2 " #2 1 (α − γ) ni α − γ SSi (g) = + ∑ . (12) 2 ni + 1 nj + 1 j∈gi∪{i}

Let (ij) ∈/ g. According to equation (12), if countries i and j establish an FTA (ij), then the change in its social surplus is computed as follows.

2 2 " #2 2 2 1 (α − γ)(ni + 1) α − γ α − γ 1 (α − γ) ni α − γ SSi (g + ij) − SSi (g) = + + − − 2 ni + 2 ni + 2 nj + 2 2 ni + 1 ni + 1 2 " #2 (α − γ) 2n2 − 5 α − γ = i + . 2 2 + 2 (ni + 1) (ni + 2) nj 2

The above equation implies that a country has strict incentive to establish more FTAs if it have established at least one FTA. Let (ij) ∈ g. If country i terminates FTA (ij), then the change in its social surplus is similarly computed as follows.

2 2 " #2 (α − γ) −2n + 4ni + 3 α − γ SS (g − ij) − SS (g) = i − . i i 2 2 n + 1 2ni (ni + 1) j

Note that the above expression is strictly negative if ni ≥ 3. These observations tell us that a country with one FTA will be strictly better off by either forming one more FTA or terminating the existing one.

4.4.2 R-Stochastically stable network

Following Goyal and Joshi (2006), we employ the notion of pairwise stable network as the solution concept. We write that network g is an equilibrium network if the strategy proﬁle constituting g is a pairwise stable equilibrium. The following proposition is due to Goyal and Joshi (2006).

Proposition 7. An equilibrium trading network is either a complete network or consists of two components; one component is n − 1 countries and is complete, and the other component has a single country.

There are two sets of pairwise stable networks. The ﬁrst set only includes a complete network, while the other set has almost complete networks in which all n − 1 countries are linked with each other. Now, we discuss the R-stochastically stable networks.

Lemma 3. Suppose that (ij) ∈ g. If ni > nj, then SSi (g) − SSi (g − ij) > SSj (g) − SSj (g − ij) .

Lemma 3 tells us that the marginal beneﬁt of an FTA is greater for the country who has more FTAs than the other country has. Applying our framework to the FTA formation game, we characterize the R-stochastically stable network as follows.

9Equivalently, one can assume that tariffs are sufﬁciently high such that no ﬁrm has an incentive to sell in another country without a tariff-free access.

19 Proposition 8. For any n ≥ 2, unique R-stochastically stable network is a complete network.

Proof of proposition 8. Observe that only stable networks are absorbing states. According to Propo- sition 7, we can restrict our attention to the transition waste between a complete network, denoted by gC, and an almost complete network, denoted by gAC. We will prove the claim by computing CR gC and R gC, and applying the Radius-Coradius theorem. Consider the transition from gAC to gC. Without loss of generality, let county 1 have no FTA (due to symmetry assumption, the transition waste will not differ). To move the state toward gC, country 1 must establish an FTA with another country j. Note that there is no waste for country j. As we discussed that a country with at least one FTA has strict incentive to establish more, the state moves to gC with waste zero after country 1 establishes an FTA with j. Thus, the waste from gAC to gC is given by

AC C AC AC W q g , g , ρ = SS1 g − SS1 g + 1j (13) 1 (α − γ) 2 α − γ 2 1 2 (α − γ) 2 α − γ 2 α − γ 2 = + − − − 2 2 2 2 3 3 n + 1 " # 1 1 = (α − γ)2 − . 24 (n + 1)2

If n < 4, the above expression is strictly negative. It implies that gAC is not an equilibrium network and gC is unique R-stochastically stable state if n < 4. For n ≥ 4, gAC is core stable and the above expression suggests that " # 1 1 (α − γ)2 CR gC = (α − γ)2 − < for n ≥ 4. 24 (n + 1)2 24

In the transition from gC to gAC, there are three sets of paths; (i) country 1 severs a link with every other (n − 2) country, (ii) each of other countries severs their link with country 1, and (iii) any mix of (i) and (ii). According to lemma 3, the waste achieves its minimum in case (i). The waste to sever a ﬁrst link is given by

C C C C W q g , g − 1j , ρ = SS1 g − SS1 g − 1j 1 (α − γ) n 2 α − γ 2 1 (α − γ)(n − 1) 2 α − γ 2 = + 2 − − 2 n + 1 n + 1 2 n n (α − γ)2 (2n − 3)(2n + 1) = . 2n2 (n + 1)2

The radius of gC is the waste that country 1 severs a link with every other (n − 2) country and is given by

" n 2 # 2 2n − 4ni − 3 α − γ R gC = W q gC, gAC , ρ = (α − γ)2 i + (n − 2) . (14) ∑ 2 ( + )2 n + 1 ni=3 2ni ni 1

20 For n = 4, a direct calculation shows that CR gC < R gC. For n ≥ 5, observe that

" n 2 # 2 2n − 4ni − 3 α − γ R gC = (α − γ)2 i + (n − 2) ∑ 2 ( + )2 n + 1 ni=3 2ni ni 1 " n−6 2 # 2 1 2n − 4ni − 3 α − γ > (α − γ)2 + (α − γ)2 i + (n − 2) 24 ∑ 2 ( + )2 n + 1 ni=6 2ni ni 1 (α − γ)2 > , 24

y where ∑n=x = 0 if x > y. The last observation together with the Radius-Coradius theorem implies that stochastic potential of gC is unique stochastically stable state for n ≥ 4. This completes the proof.

4.4.3 Asymmetric market sizes across countries

In the previous section, we assume that all countries are symmetric. Our framework can be also applied to asymmetric cases. We parameterize country size in terms of the value of α. Without loss

of generality, let α1 > α2 > ... > αn. Given the set of countries H, let H (i) denote the i-th largest country in H. Observe that the changes in social surplus by establishing an FTA and terminating an FTA are computed respectively as follows:

2 2 " #2 (αi − γ) 2n − 5 αj − γ SS (g + ij) − SS (g) = i + , (15) i i 2 2 + 2 (ni + 1) (ni + 2) nj 2

2 2 " #2 (αi − γ) −2n + 4ni + 3 αj − γ SS (g − ij) − SS (g) = i − . (16) i i 2 2 n + 1 2ni (ni + 1) j We have similar observations to the symmetric case. A country with at least one FTA has strictly positive incentive to establish more FTA regardless of the partner’s market size. And a country with at least two FTAs has no incentive to terminate any of them. The major difference from the symmetric case is that the change in the producer surplus depends on not only the number of FTAs the partner country has, but also the market size of the partner country. We have the following proposition with regards to equilibrium networks.

Proposition 9. For n ≥ 3, an equilibrium trading network is

1. a complete network,

2. or consists of k + 1 components for some k ≥ 1; one component has n − k countries and is complete, and each of the other k components has a single country such that

r r !2 r !k−1 2 2 2 α − γ ≥ 2 α − γ ≥ 2 α − γ ≥ ... ≥ 2 α − γ , H(1) 3 H(2) 3 H(3) 3 H(k) (17)

21 and αH(i) ≥ αH for all i ∈ {1, . . . , k} where H denotes the set of countries which has no FTA and √ 2 6 α = α − γ + γ. (18) H n − k + 2 (N\H)(1)

The following proposition fully characterizes the sufﬁcient condition under which a complete network is stochastically stable.

Proposition 10. Suppose that conditions (17) and (18) are strict. A complete network is unique R- stochastically stable network if n ≥ 5.

The sufﬁcient condition n ≥ 5 is actually tight. We can ﬁnd an example with n = 4 in which a complete network is not stochastically stable

Example 6. Suppose that n = 4 and that α1 = 3, α2 = α3 = α4 = 1 and γ = 0. There are two stable networks, such as a complete network and an almost complete one in which country 1 is isolated. In this economy, the latter network, gH with H = {1} is stochastically stable. To see this, observe that

37 8 W q∗ g{1}, gC = SS g{1} − SS g{1} + (1, 2) = > , 1 1 100 25 and

4 n2 − 4n − 3 α − γ 2 8 37 ∗−1 {1} C = ( − )2 1 1 + i = < W q g , g α1 γ ∑ 2 ∑ . 2 ( + ) ni + 1 25 100 n1=3 2n1 n1 1 i∈{3,4}

{1} 4 {1} 4 {1} Note that the global social surplus is not maximized in g , i.e. ∑i=1 Si g < ∑i=1 Si g . However, 147 SS g{1} = 6 > = SS gC . 1 25 1

5 Discussion

In many of the applications in Section 4, our model selects Pareto efficient strategy profiles. One might tend to conclude that these results are straight-forward because stochastic potentials are computed based on the sum of payoff decreases between strategy profiles. However, Pareto efficiency of resulting outcomes is not always guaranteed. Moreover, larger sets of feasible coalitions may not always result in more efficient outcomes. These points are illustrated in the following example.

Example 7. Consider the following three-player normal form game G.

A3 B3 2 2

A2 B2 A2 B2 1 A1 4, 4, 4 0, 0, 0 A1 0, 0, 0 0, 3, 6 A2 0, 0, 0 3, 6, 0 A2 6, 0, 3 2, 2, 2

22 Suppose that interaction I is equivalent to G, i.e. S = S. Let A = (A1, A2, A3) and B = (B1, B2, B3). Observe that the sets of Nash equilibria and pairwise stable equilibria coincide and are given by {A, B}, and that A Pareto dominates B. A is the unique R-stochastically stable strategy proﬁle when R = N, but B is the unique R-stochastically stable proﬁle when R = {J ⊂ N : |J| ≤ 2}.

It might seem puzzling that the Pareto dominated equilibrium is selected when pairs of players can cooperate. However, if some coalitions make it less costly to escape from a Pareto efficient equilibrium, then allowing such coalitions to be feasible may lead to inefficient play. In Example 7, any unilateral deviation from strategy profile A costs 4, while any unilateral deviation from B costs 2. In contrast, any pairwise deviation from A, (B2, B3) for example, costs 1 and any pairwise deviation in B costs 4. Taking the minimum costs of deviating from each equilibria, we easily find that B is R- stochastically stable with R = {J ⊂ N : |J| ≤ 2} although B is Pareto dominated. This example also illustrates that a larger set of feasible coalitions may not necessarily lead to more efficient outcomes because players may cooperate to deviate from efficient outcomes.

6 Conclusion

We have extended the stochastic stability approach to interactions with joint feasibility constraints on strategy profiles, and in which agents may form coalitions. We also introduce a flexible coalitional solution concept R-stable equilibrium. Our main result has presented a characterization of R-stochastically stable outcomes under the logit-response dynamic. Our model is a general framework in the sense that it handles a broader class of settings and an arbitrary set of coaltions; the modeller chooses an interaction and a set of admissible coalitions, denoted by R, and then our result provides the R-stochastically stable outcomes of the interaction. When an interaction is R-acyclic, a subset of R-stable equilibria is selected. We have defined coalitional potential and shown that R-stochastically stable states are coalitional potential maximizers if the interaction exhibits a coalitional potential. Applying our version of the Radius-Coradius theorem, we also have shown that a complete network is R-stochastically stable in the network formaions of bilateral trade agreements. Many questions remain to be answered. One is whether the stochastically stable strategy profiles can be characterized succinctly in matching problems, such as Gale-Shapley marriage problems and hospital-intern problems. Jackson and Watts (2002) showed that all stable matchings in these problems are stochastically stable under the best response with mutations dynamic, under which all mistakes are equally likely. The result will likely differ under the logit choice rule; our model may select a proper subset of stable matchings. A second question is whether our model can be extended in a way to examine stochastic stability of coalition-proof equilibrium due to Bernheim et al. (1987). Although we have shown that our model can examine stochastic stability of various coalitional solution concepts, our model may not be suitable for coalition-proof equilibrium. Coalition-proof equilibrium is robust against self-enforcing joint deviations by coalitions. Roughly describing, a joint deviation by coalition J is self-enforcing if no proper subset of J, taking the actions of its complement as fixed, can agree to deviate in a way that makes all of its members better off. In the unperturbed dynamic of our model, a coalition- proof equilibrium is not necessarily absorbing because it may be upset by some non self-enforcing joint deviations. This suggests that the predictions of our model might differ from any subset of

23 coalition-proof equilibria. An interesting problem in further study is to investigate what types of differences might emerge, and characterize a class of interactions in which our predictions coincide with some subset of coalition-proof equilibria.

7 Appendix

7.1 Proofs of theorems in Section 3

This follows the proofs of Lemma 2 and 3 in Alos-Ferrer´ and Netzer (2010). Let T (s) denote the set of all revision s-trees. Let Sxx denote the set of x-tuple of possible strategy profiles of x agents. n o JJ ⊂ nn ( 00) ∈ 0 ∈ Let Qs,s0 S denote the set of all n-tuple of strategy profiles such that, s j sj, sj for j J ( 00) = ∈ 00 ∈ JJ 10 ⊂ nn × nn × and s j sj for j / J if s Qs,s0 . Let L S S R denote the set of pairs of states and one revising set such that, for all (s, s0, J) ∈ L, transition (s, s0) is feasible when the revising set is R 0 ⊆ 0 → nn ( 0 ) ∈ JJ J. A virtual realization for L L is a mapping r : L S such that r s, s , J Qs,s0 for all (s, s0, J) ∈ L0. A complete realization is a virtual realization for L.A completion of revision tree (T, ρ) is a complete virtual realization such that r (s, s0, ρ (s, s0)) = (s0)n for all (s, s0) ∈ T. Let C (T, ρ) be the set of all completions of (T, ρ). Let Nρ = {(s, s0, ρ (s, s0)) |(s, s0) ∈ T } be the set of two states and revising set induced by ρ for tree T. Let R (Nρ) denote the set of virtual realizations of the selection ρ. Also define 2 0 S f = {(s, s ) ∈ S × S : Rs,s0 6= φ}.

2 S f denotes the set of pairs of states such that a transition between given two states is feasible. Lemma 4. The stationary distribution πη satisﬁes for each s ∈ S

−1 πη (s) ∝ ∑ P (T, ρ) ∑ eη Q(r), (T,ρ)∈T (s) r∈C(T,ρ) where 0 P (T, ρ) = ∏ qρ(s,s0)qs0 ρ s, s , s (s,s0)∈T and Q (r) = ∑ ∑ UJJ (r (s, J) , s) 0 2 J∈R 0 (s,s )∈S f s,s where 0 = 0 0 ∈ nn ∈ UJJ s , s ∑ uj s j , s−J for s S and s S . j∈J Proof. By Freidlin and Wentzell (1988), we know that

η ( ) η π s ∝ ∑ ∏ Ps,s0 . T∈T (s) (s,s0)∈T

10 0 0 0 For example, suppose n = 2, J = {2} and Si = si, si be the set of agent i strategy. Letting v = {s1, s2} and s = {s1, s2}, JJ Qs,s0 is given by JJ = {( ) ( )} ( ) 0 Qs,s0 s1, s2 , s1, s2 , s1, s2 , s1, s2 .

24 Note that s-trees including infeasible transitions contribute zero to the sum above. Using equation (4), decompose the transition probabilities as follows.

 h i  exp η−1u s0 , s η j J −J P = q q 0 (J, s) ∏ s,s0 ∏  ∑ J s ∏ h i  0 0 ∈ −1 0 −1 (s,s )∈T (s,s )∈T J Rs,s0 j∈J exp η uj sJ, s−J + exp η uj (s)  h i  exp η−1 u s0 , s  ∑j∈J j J −J  = q q 0 (J, s) , ∏  ∑ J s  0 J∈R 0 −1 00  (s,s )∈T s,s JJ u s s ∑s00∈Q exp η ∑j∈J j J , −J J s,s0 j

00 00 JJ where s is the j-th component of s ∈ Q 0 . Let S (T) denote the set of all revision selections ρ J j J s,s for tree T. Then, the transition probabilities further can be written as   h −1 0 i exp η ∑ u s , s− η  j∈Jρ j J Jρ  P =  q q 0 J , s  ∏ s,s0 ∑  ∏ Jρ s ρ  (s,s0)∈T ρ∈S(T) (s,s0)∈T −1 00  ∑ Jρ Jρ exp η ∑ ∈ uj s , s−J s00∈Q j Jρ J ρ J s,s0 j   h − 0 i 1 0  exp η ∑(s,s )∈T UJρ (s , s)  = P (T, ρ) h i  , ∑  −1 00  ρ∈S(T)  ∑ Jρ Jρ exp η UJ J s , s  ∏ s00∈Q ρ ρ J (s,s0)∈T J s,s0

0 0 where Jρ = ρ (s, s ) for (s, s ) ∈ T. Expanding the denominator in the last expression yields   h i −1 00 = −1 0 0 ∏ ∑ exp η UJρ Jρ sJ , s ∑ exp η ∑ UJρ Jρ r s, s , ρ s, s , s  . ( 0)∈ Jρ Jρ ∈ ( ρ) ( 0)∈ s,s T s00∈Q r R N s,s T J s,s0

Multiplying and divide the last expression of the transition probabilities by   −1 0 ∑ exp η ∑ UJJ r s, s , J , s  , r∈R(L\Nρ) (s,s0,J)∈L\Nρ

we obtain     η = ( ) −1 0 × ∏ Ps,s0 ∑ P T, ρ  ∑ exp η ∑ UJJ r s, s , J , s  (s,s0)∈T ρ∈S(T) r∈C(T,ρ) (s,s0 J)∈L   −1 −1 0   ∑ exp η ∑ UJJ r s, s , J , s   . r∈R(L) (s,s0,J)∈L

The last term in brackets is independent of T and ρ, and hence it is irrelevant for proportionality of πη (s).

The existence of the limiting stationary distribution is provided by the lemma below.

25 η Lemma 5. The limiting stationary distribution limη→0 π exists.

Proof of Lemma 5. Deﬁne Λ as

Λ ≡ max max max Q (r) . s∈S (T,ρ)∈T (s) r∈C(T,ρ)

η It is obvious that limη→0 π (s) = 0 for any state s ∈ S for which

max max Q (r) < Λ. (T,ρ)∈T (s) r∈C(T,ρ)

Thus, state s ∈ S is stochastically stable only if

max max Q (r) = Λ. (T,ρ)∈T (s) r∈C(T,ρ)

Proof of Theorem 1. For any revision tree (T, ρ), the completion which maximizes Q (r) among all r ∈ C (T, ρ) must involve realization rmax 0 such that

0 max0 0 0 ρ max u s , s− − u r s, s , J , s− = 0 ∀j ∈ J and ∀ s, s , J ∈/ N . 0 j J J j J sJ ∈SJ

Since C (T, ρ) includes all completion of (T, ρ), such rmax 0 exists. Now, deﬁne rmax ∈ R (L) such that

0 max 0 0 max u s , s− − u r s, s , J , s− = 0 ∀j ∈ J and ∀ s, s , J ∈ L. 0 j J J j J sJ ∈SJ

Then, a state s ∈ S maximizes max(T,ρ)∈T (s) maxr∈C(T,ρ) Q (r) if and only if it maximizes

0 max 0 max UJρ s , s − UJρ Jρ r s, s , Jρ , s (19) (T )∈T (s) ∑ ,ρ (s,s0)∈T

0 0 where Jρ = ρ (s, s ) for (s, s ) ∈ T. Noting that the summand in (19) can be written as n o U s0 s − U rmax s s0 J s = u s0 s − u s0 s u (s) Jρ , Jρ Jρ , , ρ , ∑ j Jρ , −Jρ max j Jρ , −Jρ , j , j∈Jρ

we ﬁnd that the sum in equation (19) is equivalent to −W (T, ρ). Then, the claim follows.

Proof of Theorem 2. Let U denote the set of states that correspond to a R-stable equilibrium. If a state s ∈ S does not correspond to a R-stable equilibrium, there must exist an improving path from s by definition. Since there is no closed cycle in (I, R), this path will lead to a state s0 that is R-stable equilibrium. This implies that the coradius of U (see equation (9)) satisfies that CR(U) = 0. Suppose that there exist some u1 ∈ U such that the radius of u1 (see equation (6)) is R(u1) = 0. Then, there exists an improving path that starts from u1. Due to the acyclic condition, the improving path from 0 0 0 u1 must lead to some u ∈ U. Let U = U \{u1}. If there exists u2 ∈ U such that R(u2) = 0, then 0 0 0 0 0 refine U by letting U = U \{u1, u2}. We continue this discussion until R(u ) > 0 for all u ∈ U .

26 Note that, by construction, CR(U0) = 0 holds and the radius of U0 satisﬁes that R(U0) > 0. The claim follows by theorem 2.

∗ ∗ Proof of Theorem 2. Let U2 = U \ U1. Choose u2 ∈ U2. For any state u, let T (u) and ρ (u) be a revision tree of u such that W (T∗ (u) , ρ∗ (u)) = W (u). Also let d∗ (u, u0) ⊂ T∗ (u0) be a path from 0 ∗ 0 0 0 0 state u to u on tree T (u ). Choose u ∈ U1 such that, for some (d , ρ) ∈ D(u2, u ),

0 0 W d u2, u , ρ ≤ CR (U1) .

By definition of the coradius, such u0, d0 and ρ must exist. ∗ 0 ∗ ∗ 0 ∗ Now suppose d (u , u2). Let u1 ∈ U1 such that d (u1, u2) ⊆ d (u , u2) and d (u1, u2) does not ∗ 0 pass any u ∈ U1 except u1. In words, u1 is the last state in U1 that appears in d (u , u2). Similarly, ∗ ∗ ∗ let u3 be such that d (u1, u3) ⊆ d (u1, u2) and u3 is the first state in d (u1, u2) that is not in B (u1). According to the definition of the radius, we know that

∗ W (d (u1, u3)) ≥ R (U1) .

Now, delete the part of the path from u1 to u3, and then make edges from disconnected states to u1. Since all disconnected states are in B (u1), we can make a path to u1 with waste zero. This will 0 0 reduce the waste by weakly greater than R (U1). Then, add d (u2, u ) to the revision tree. Delete any 0 ∗ duplicated transition when adding d . Delete edges belonging to T (u2) if there are more than one edge starting from the same state. This will increase the waste by weakly less than CR (U1). After these two operations, observe that we have constructed a revision tree of u1. By construction, we know that

W (u1) ≤ W (u2) − R (U1) + CR (U1) < W (u2) .

According to theorem 1, u2 is not stochastically stable. Since the choice of u2 is arbitrary, any u ∈ U2 is not stochastically stable. The claim follows from the existence of stochastically stable state.

7.2 Proofs of theorems in Section 4

∗ Proof of Proposition 3. Let bs denote a state that does not maximize P. Also let s denote a state maximizing P. Let Tb and ρb be a bs-tree and a revision selection such that W Tb, ρb = W (bs) .

∗ ∗ 0 00 0 00 ∗ Let q (s , bs) = {(s , s1) , (s1, s2) ,..., (sh, bs)} such that (s , s ) ∈ Tb for all (s , s ) ∈ Q (bs, s ). In words, ∗ ∗ ∗ q (s , bs) is a path from s to bs along tree Tb. Since Tb is a tree, such a path exists and is unique. Let T is a s∗-tree such that 00 0 ∗ 0 00 ∗ s , s ∈ T for all s , s ∈ q (s , bs) , and 0 00 ∗ 0 00 ∗ s , s ∈ T for all s , s ∈/ q (s , bs) .

27 Also let ρ∗ be a revision selection such that

∗ 00 0 0 00 0 00 ∗ ρ s , s = ρb s , s for all s , s ∈ q (s , bs) ,

and ∗ 0 00 0 00 0 00 ∗ ρ s , s = ρb s , s for all s , s ∈/ q (s , bs) . (20) Observe that   ∗ ∗ 00 0 0 00 W (T , ρ ) = W Tb, ρb + ∑  ∑ max uj s − uj s , 0 − ∑ max uj s − uj s , 0  . 0 00 ∗ ∗ 00 0 0 00 (s ,s )∈q(s ,bs) j∈ρ (s ,s ) j∈ρb(s ,s )

According to equality (20), we obtain

∗ ∗ 00 0 0 00 W (T , ρ ) = W Tb, ρb + ∑ ∑ max uj s − uj s , 0 − max uj s − uj s , 0 . 0 00 ∗ 0 00 (s ,s )∈q(s ,bs) j∈ρb(s ,s ) (21) 00 0 0 00 Now, noting that uj (s ) − uj (s ) = 0 for j ∈/ ρ (s , s ), we can rewrite the right most sum in the last expression as

00 0 0 00 ∑ max uj s − uj s , 0 − max uj s − uj s , 0 j∈ρ(s0,s00) 00 0 0 00 = ∑ max uj s − uj s , 0 − max uj s − uj s , 0 j∈N = P s00 − P s0 .

Then, we can rewrite the equation (21) as

∗ ∗ 00 0 W (T , ρ ) = W Tb, ρb + ∑ P s − P s 0 00 ∗ (s ,s )∈Q(s ,bs) ∗ = W Tb, ρb + P (bs) − P (s ) < W Tb, ρb .

∗ The last inequality comes from the fact that bs does not maximize the potential, but s does. Thus, a state has the lowest stochastic potential if and only if it maximizes the coalitional potential . The claim follows from Theorem 1.

Proofs for Section 4.2 Proof of Proposition 4. Recall that S = s ∈ S si 6= sj ∀i, j ∈ N . Note that the interaction of the house exchange economy admits a coalitional potential function P over S such that for all s ∈ S

P (s) = ∑ ui (s) . i∈N

The claim follows from Proposition 3.

28 Proofs for Section 4.3

Proof of Lemma 2. First, observe that any strategy proﬁle s is a core stable equilibrium with respect to R. Next, note that there always exists at least one agent who will be worse off by transition (s, s0) for any s, s0 ∈ S . It implies that the waste of any path (s, s0) is strictly positive. Then, the claim follows.

Proof of Corollary 3. Suppose that agent 1 is in the center. Note that any state in which agent 1’s contributions to two links differ by strictly greater than γ cannot be stochastically stable due to Propo- sition 6 and symmetry of peripheral agents. Then, a stochastically stable state must be a state such that agent 1 contributes kγ to each link of m agents for some m, k ∈ N and (k + 1) γ to each link of remaining peripheral agents. Without loss of generality, let agent 1 contribute kγ to links to agents 2, . . . , m + 1. Observe that

Y1 (s) = (n − 1) δ − (mkγ + (n − 1 − m)(k + 1) γ) ,

2 Yi (s) = δ + (n − 2) δ − (c − kγ) for 2 ≤ i ≤ m + 1,

2 Yi (s) = δ + (n − 2) δ − (c − (k + 1) γ) for m + 2 ≤ i ≤ n.

In a stochastically stable state, (m, k) must be such that

Y1 (s) − Yi (s) ≤ γ for 2 ≤ i ≤ m + 1,

Yi (s) − Y1 (s) ≤ γ for m + 2 ≤ i ≤ n.

These two inequalities are reduced to (n − 2) δ − δ2 + c ≤ γ (nk + n − m) ≤ (n − 2) δ − δ2 + c + γ. (22)

According to Corollary (1), k∗ is uniquely determined by the following inequalities: γnk∗ ≤ (n − 2) δ − δ2 + c < γn (k∗ + 1) (n − 2) δ − δ2 + c (n − 2) δ − δ2 + c ⇔ − 1 < k∗ ≤ . γn γn

Note that k∗ ≥ 0 due to that RHS of the above inequality is strictly positive. Also note that such k∗ ≤ c/γ exists because RHS is less than c/γ. Then, m∗ of a stochastically stable state is such that (k∗, m∗) maximizes P. Note that k∗ is unique, but m∗ may or may not be unique.

Proofs for Section 4.4

Proof of lemma 3. The proof is straightforward. The changes in two countries’ social surplus are respectively computed as

" 2 # " #2 2n − 4ni − 3 α − γ SS (g) − SS (g − ij) = (α − γ)2 i + , i i 2 2 n + 1 2ni (ni + 1) j

29   2 2 2nj − 4nj − 3 α − γ ( ) − ( − ) = ( − )2 + SSj g SSj g ij α γ  2  . 2 ni + 1 2nj nj + 1 Observe that   2 (α − γ) 4nj + 3 4ni + 3 SSi (g) − SSi (g − ij) − SSj (g) − SSj (g − ij) =  −  > 0. 2 2 2 2 ( + )2 nj nj + 1 ni ni 1

For the last inequality to hold, note that ! ∂ 4x + 3 −12x2 − 16x − 6 = < 0 for x > 0. ∂x x2 (x + 1)2 x3 (x + 1)3

Proof of proposition 9. First, equation (15) shows that country i with ni ≥ 2 has strict incentive to form more. It implies that a stable network has no two component each of which consists of two or more

countries. For ni ≥ 3, equation (16) is strictly negative. This implies that a complete network is stable. For the second case, we ﬁrst show that a country in H, denoted by H (i), has no strict incentive to establish an FTA with another country in H, denoted by H (j), such that i < j. Since H (i) and H (j) have no FTA, note that nH(i) = nH(j) = 1. To see that H (i) has no incentive to form a link with H (j), observe that

2 2 αH(i) − γ αH(j) − γ SS (g + (H (i) , H (j))) − SS (g) = − + H(i) H(i) 24 32 2 2 8 3 αH(i+1) − γ αH(j) − γ ≤ − + ≤ 0 24 9

where the ﬁrst weak inequality comes from condition (17). Next, we show that a country in H, denoted by H (i), has no strict incentive to establish an FTA with any country in N \ H. Let j denote

a country in N \ H. Note that nj = n − k, since the network is complete in N \ H. Observe that

2 2 αH(i) − γ αH(j) − γ SSH(i) (g + (H (i) , j)) − SSH(i) (g) = − + 24 (n − k + 2)2 2 2 24 α(N\H)(1) − γ αH(j) − γ ≤ − + ≤ 0 (n − k + 2)2 · 24 (n − k + 2)2

where the ﬁrst weak inequality comes from condition (18). Thus, a network consisting of k + 1 components and satisfying conditions (17) and (18) is stable. Finally, according to the above compu- tations, observe that if either of (17) or (18) is violated, there exists a country in H, denoted by H (i), which has strict incentive to establish an FTA with H (i + 1) or some country in N \ H. Observe that country H (i + 1) always has strict incentive to establish an FTA with H (i). Any country in N \ H also always has strict incentive to establish an FTA with any country according to (15). Thus, any

30 other network that does not satisfy hypotheses is not stable.

The following lemma will be useful to prove the main result.

Lemma 6. Suppose that the network is complete among members in K ⊆ N. If |K| ≥ 5, then the least waste 2 for any country i ∈ K to be isolated from other countries is strictly greater than (αi − γ) /24.

Proof of lemma 6. Let K0 = K \{i}. There are three possible cases to sever all links from country i; (i) country i to sever all its links, (ii) other countries to server their link with i or (iii) any mixture of (i) and (ii). Let W denote the waste for country i to be isolated. case (i): The waste that country i severs a link with every other country is bounded below by

" # " #2 5 2n2 − 4n − 3 α − γ ≥ ( − )2 i i + j W αi γ ∑ 2 ∑ . 2 ( + ) nj + 1 ni=3 2ni ni 1

case (ii): Suppose that country j severs a link with i when ni = 3. Then, the waste is given by   2n2 − 4n − 3 2 2 2 j j αi − γ (αi − γ) SSj (g) − SSj (g − ij) = αj − γ   + > . 2 2 4 24 2nj nj + 1

case (iii): The observation in case (ii) implies that we only need to consider cases that country i

severs a link when ni = 3. If some country j severs a link with i when ni = 4, then the waste is bounded below by

" 2 # 2 2 2 · 3 − 4 · 3 − 3 αi − γ W > (αi − γ) + 2 · 32 (3 + 1)2 5 1.2 > (α − γ)2 . i 24

If i severs a link when ni = 3, 4 and another country j severs a link with i when ni = 5, then the waste can be written as

" 4 2 # 2 2 2ni − 4ni − 3 αi − γ W > (αi − γ) + ∑ 2 ( + )2 6 ni=3 2ni ni 1 1.3 > (α − γ)2 . i 24

If i severs a link when ni = 3, 5 and another country j severs a link with i when ni = 4, then the waste can be written as   2 2 2 2ni − 4ni − 3 αi − γ W > (αi − γ)  ∑  + n2 (n + )2 5 ni∈{3,5} 2 i i 1 1.5 > (α − γ)2 . i 24

2 According to above observations, the waste is strictly greater than (αi − γ) /24 in any possible transition. This completes the proof.

31 The following lemma considers the case n = 5.

Lemma 7. Suppose asymmetric market size across countries and that conditions (17) and (18) are strict. A complete network is stochastically stable if n = 5.

H Proof of lemma 7. In the following discussion, note that nj ≤ 4 ∀j ∈ N if |H| ≥ 1. For g to be stable, the following inequality must hold: √ 6 α − γ > (α − γ) . (23) H(1) 3 1

There are two possible cases; (i) only country 1 satisﬁes condition (23), and (ii) there are more than one country which satisfy (23).

Suppose case (i). In this case, observe that Si (g + 1i) − Si (g) > 0 always holds for any i 6= 1 and g such that (1i) ∈/ g. This observation implies that 1 ∈ H for any stable network gH, i.e. country 1 must be isolated. Otherwise, any isolated country j forms an FTA with country 1 which also has an incentive to do so. According to lemma 6, the minimum waste for the transition from gC to any stable network gH is bounded by

(α − γ)2 W q gC, gH , ρ > 1 . 24

This inequality implies that (α − γ)2 R gC > 1 . 24 For any stable network gH, a transition from gH to gC requires country 1 to establish an FTA. It incurs the waste given by, for some country j 6= 1,

2 " #2 2 (α − γ) αj − γ (α − γ) W q gH, gC , ρ = 1 − < 1 . 24 nj + 2 24

This implies that (α − γ)2 CR gC < 1 . (24) 24 By the Radius-Coradius theorem, gC is unique stochastically stable state in case (i). Now, suppose case (ii) that there exists country j 6= 1 satisﬁes condition (23). Choose a stable network gH with |H| ≥ 1. Lemma 6 tells us that the minimum waste for the transition from gC to gH is bounded by

2 − " #2 αH(1) γ αj − γ W q gC, gH , ρ ≥ + 24 ∑ n + 2 j6=H(1),(N\H)(1) j (α − γ)2 > 1 , 36

where the second inequality comes from condition (23). Since the choice of gH is arbitrary, the above

32 inequality implies that (α − γ)2 R gC > 1 . 36 For any gH with 1 ∈ H, observe that the state will move to gC with waste zero if country 1 establishes an FTA. The minimum waste of a transition from gH to gC is given by

(α − γ)2 α − γ 2 (α − γ)2 α − γ 2 W q gH, gC , ρ = 1 − 2 ≤ 1 − 2 24 n2 + 2 24 6 (α − γ)2 6 (α − γ)2 < 1 − 9 1 24 36 5 (α − γ)2 (α − γ)2 = 1 < 1 . 216 36

Similarly, for any gH with 1∈ / H, the state will move to gC with waste zero if country H (1) establishes an FTA. The transition from gH to gC incurs the waste given by,

2 αH(1) − γ α − γ 2 W q gH, gC , ρ = − 1 24 n1 + 2 (α − γ)2 (α − γ)2 (α − γ)2 (α − γ)2 < 1 − 1 = 1 < 1 . 24 36 72 36

The above two inequalities imply that

(α − γ)2 CR gC < 1 . 36

Again, according to the Radius-Coradius theorem, gC is unique stochastically stable state in the case that more than one country satisfy condition (23). This completes the proof.

Proof of proposition 10. Note that any absorbing state is singleton when (17) and (18) are strict. With lemma 7, we need to prove the claim only for n ≥ 6. The plan of the proof is that I will ﬁrst show the minimum waste for the transition from any stable network gH 6= gC to a complete network gC. Then, I will show that any stable network with n − |H| ≥ 5 is not stochastically stable. Finally, I will address remaining stable networks by applying the Radius-Coradius theorem. Let gH be a stable network such that H denotes the set of countries that have no FTA. Let K = N \ H. The minimum waste from gH to gC can be written as, for some ρ, ( SS gH + H (1) K (1) − SS gH for |H| = 1, H C = H(1) H(1) W q g , g , ρ H H (25) SSH(1) g + H (1) H (2) − SSH(1) g for |H| ≥ 2.

To see this, suppose |H| = 1 ﬁrst. Then, it is obvious that i ∈ H must establish an FTA with j ∈ K to move the state toward gC. Then, it is least costly for i to form an FTA with K (1) due to the largest beneﬁt from exporting. For |H| ≥ 2, observe that the least cost FTA for H (1) to form is the FTA with H (2) due to condition (17) and (18). Suppose that the state starts from gH and H (1) has established an FTA with H (2). Let gH−1,2 denote the resulting network. Note that country H (1) has strict

33 incentive to form an FTA with any country. Then, for any i ≥ 2, observe that

2 2 αH(i) − γ αH(1) − γ H−1,2 H−1,2 SSH(i) g + H (1) H (i) − SSH(i) g = − + 24 (2 + 1)2 2 2 8 αH(i) − γ 3 αH(i) − γ > − + = 0. 24 (2 + 1)2

The ﬁrst strict inequality comes from condition (17). The above inequality shows that H (1) and H (3) will form an FTA with waste zero. A similar computation will show that H (2) and H (4) will form an FTA with waste zero and further show that all countries in H will form an FTA with waste zero after H (1) forms one. Then, the state moves toward gC with waste zero. Thus, the minimum waste is given by equation (25). Suppose a stable network gH such that |H| ≥ 1 and n − |H| ≥ 4. Let T∗ gH and ρ∗ gH be a revision tree of gH that minimizes W gH. Let q∗ g, gH ⊂ T∗ gH be a path from g to gH along T∗ gH. Let H − h = H \{h} for h ∈ H. And let gH−h = gC if H = {h}. Choose h ∈ H such that gH−h is stable and q∗ gH−h, gH ⊂ T∗ gH does not pass any other stable network than gH−h. Since 0 path q∗ gC, gH must pass some stable network gH such that |H0| = |H| − 1, such country h exists. Now, consider the transition from gH−h to gH There are three cases for the transition from gH−h to gH; (i) country H (h) to sever all its links, (ii) other countries in K to server their link with H (h) or (iii) any mixture of (i) and (ii). Note that, to ensure the least cost transition, the link between H (h) and K (1) must be severed by H (h) when nH(h) = 2, because equation (16) tells us that severing a link with a larger country will generate a greater waste compared to a smaller country when ni ≥ 3. Lemma 6 tells us that 2 αH(h) − γ W q∗ gH−h, gH , ρ∗ > . 24 According to the above discussion, gH + (H (h) , K (1)) , gH must be the last component of q∗ gH−h, gH . Construct the reversed path q∗−1 gH−h, gH . Note that q∗−1 gH−h, gH is feasible with some appropriate ρ. The waste of the reversed path is the waste of transition gH + (H (h) , K (1)) , gH. Observe that

∗−1 H−h H H H W q g , g , ρ = SS1 g − SS1 g + (H (h) , K (1)) 2 αH(h) − γ α − γ 2 = − K(1) 24 |K| + 1 < W q∗ gH−h, gH , ρ∗ .

By lemma 8, gH is not stochastically stable. Now, we consider cases that n − |H| ≤ 3. Since n ≥ 6, it implies that |H| ≥ 3. Observe that, for any stable gH with n − |H| ≤ 3, H must include the largest country, i.e. 1 ∈ H. To see this, suppose

34 that 1∈ / H. Then, observe that conditions (17) and (18) imply that

r r √ 2 2 2 6 α − γ ≥ 2 α − γ > 2 · α − γ (26) H(1) 3 H(2) 3 n − |H| + 2 (N\H)(1) 4 > (α − γ) . 3 1

This violates the assumption that country 1 has the largest market. Now, let U be the set of absorbing

states, or stable networks. Let U1 be the set of stable networks in which country 1 has at least one FTA. Let U2 = U \ U1. According to the above observation, note that U2 includes all stable networks gH such that n − |H| ≤ 3. In addition, the assumption that n ≥ 6 together with inequality (26) H H ensures that g ∈ U2 if n − |H| ≤ 4. This further implies that n − |H| ≥ 5 must hold for all g ∈ U1. By the deﬁnition of U1, any transition from U1 to U2 must result in country 1 being isolated. Then, lemma 6 tells us that such a transition must incur the transition waste as,

(α − γ)2 W q u, u0 , ρ > 1 for any u ∈ U and u0 ∈ U . 24 1 2

This implies that (α − γ)2 R (U ) > 1 . 1 24 H Now, suppose the transition from U2 to U1. By deﬁnition, country 1 has no FTA in any g ∈ U2. Ac- cording to the previous discussion, the state will move to gC with zero waste if country 1 establishes H an FTA with another country. Thus, the waste of the transition from U2 to U1 is, for any g ∈ U2 some j 6= 1

2 " − #2 2 0 H H (α1 − γ) αj γ (α1 − γ) W q u , u , ρ ≤ SS1 g − SS1 g + 1j = − < . 24 nj + 2 24

This implies that (α − γ)2 CR (U ) < 1 . 1 24

According to the Radius-Coradius theorem, the limiting stationary distribution places one on U1. H H C But we know that g ∈ U1 is not stochastically stable, if g 6= g . By the existence, the limiting stationary distribution places one on gC.

To prove lemma 8, we need the following notations. Let (T∗ (s) , ρ∗ (s)) be the revision tree such that it minimizes the stochastic potential of given state v. Let q∗ (s, s0) ⊂ T∗ (s0) be a path from v to s0 ∗ 0 ∗ 0 −1 along T (s ) and W (q (s, s )) its stochastic potential. Let q (s1, sk) be a reversed path of q (s1, sk) = −1 {(s1, s2) , (s2, s3) ,..., (sk−1, sk)}, that is q (s1, sk) = {(sk, sk−1) , (sk−1, sk−2) ,..., (s2, s1)}. We say 0 that path q (s, s ) is feasible if there exists ρ such that si+1 can be reached from si through coalition 0 0 0 ρ (si, si+1) for all (si, si+1) ∈ q (s, s ). We let W (q (s, s ) , ρ) take positive inﬁnity if q (s, s ) is not feasible through mapping ρ. We have the following lemma.

Lemma 8 (Reversed least-cost path lemma). Suppose that q∗−1 (s, s0) is a feasible path for s, s0 ∈ V. If W (q∗ (s, s0)) > W q∗−1 (s, s0), then s0 cannot be stochastically stable. If W (q∗ (s, s0)) = W q∗−1 (s, s0)

35 and s0 is stochastically stable, then v is also stochastically stable.

Proof. Let T∗ (s0) and ρ∗ (s0) minimize the stochastic potential of s0, i.e. W (T∗ (s0) , ρ∗) = W (s0). By deﬁnition, there must exist T∗ (s0) and ρ∗ such that T∗ (s0) include q∗ (s, s0). We choose such (T∗ (s0) , ρ∗) if it is not unique. Then, construct T (s) by

T (s) = T∗ s0 \ q∗ s, s0 ∪ q∗−1 s, s0 .

Note that T (s) is actually a tree. In addition, T∗ (s0) and T (s) coincide except q∗ (s, s0) along which the direction is opposite between two trees. By the assumption of feasibility, there exists ρ such that

ρ (x, y) = ρ∗ (x, y) for (x, y) ∈/ q∗ s, s0 ∗ 0 ρ (y, x) ∈ Ry,x for (x, y) ∈ q s, s .

Observe that W (T (s) , ρ) = W T∗ s0 \ q∗ s, s0 , ρ∗ s0 + W q∗−1 s, s0 , ρ

< W T∗ s0 \ q∗ s, s0 , ρ∗ s0 + W q∗ s, s0 , ρ∗ s0 = W T∗ s0 , ρ∗ .

It implies that s0 cannot be stochastically stable. We can prove the latter claim of the lemma in a similar discussion.

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