The Electronic Mail Game: Strategic Behavior Under" Almost Common Knowledge"

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Ariel Rubinstein

The American Economic Review, Vol. 79, No. 3. (Jun., 1989), pp. 385-391.

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http://www.jstor.org Thu Mar 27 12:47:26 2008 The Electronic Mail Game: Strategic Behavior Under "Almost Common Knowledge"

The paper addresses a paradoxical game-theoretic example which is closely related to the coordinated attack problem. Two players have to play one of two possible coordination games. Only one of them receives information about the coordination game to be played. It is shown that the situation with "almost common knowledge" is very diferent from when the coordination game played is common knowledge.

A very basic assumption in all studies (for a recent presentation of this literature of game theory is that the game is "com- see Ken Binmore and Adam Brandenberger, mon knowledge." Following John Harsanyi 1987). Intuitively speaking it is common (1967), situations without common knowl- knowledge between two players 1 and 2 that edge are analyzed by a game with incom- the played game is G, if both know that the plete information. A player's information is game is G, 1 knows that 2 knows that the characterized by his "type." Each player game is G and 2 knows that 1 knows that "knows" his own type and the prior distri- the game is G, 1knows that 2 knows that 1 bution of the types is common knowledge. knows that the game is G, and 2 knows that Jean-Francois Mertens and Samuel Zamir 1 knows that 2 knows that the game is G (1985) have shown that under quite general and so on and so on. conditions one can find type spaces large One of the main difficulties with this intu- enough to carry out Harsanyi's program and itive definition (and with the formal defini- to transform a situation without common tions which capture this perception) is that knowledge into a gaine with incomplete in- even "simple" sentences like "I do not know formation in which the different types may that you do not know that I know that you have different states of knowledge. Har- do not know that I know" are very difficult sanyi's method became the cornerstone of all to visualize, thus malung an assessment of modem analyses of strategic economic be- their validity problematic. Therefore it would havior in situations with asymmetric infor- be interesting to understand whether a mation (i.e., most of the theoretical Indus- game-theoretic informational structure, re- trial Organization literature). ferred to as "almost common knowledge," in What does it mean that the game G is which only a finite (but large) number of "common knowledge"? Following David propositions of the type "1 knows that 2 Lewis (1969), Stephen Schiffer (1972), and knows that 1 knows.. .that the game is G" Robert Aumann (1976), this concept has are true, is very different from the situation been studied thoroughly by relating it to where the game G is common knowledge. In concepts of "knowledge" and "probability" this short paper I will present a simple example of a situation with "almost common knowledge" of the game. The situation is analyzed using, as a tool, the idea of a game *Department of Economics, The Hebrew University, with incomplete information. It is shown Jerusalem, and the London School of Economics, that the game-theoretic "prediction" for the Houghton Street, London, England W02A2AE. My thanks to Ken Binmore, Edi Dekel, John Geanakopo- "almost common knowledge" situation is los, Avner Shaked, Chuck Wilson, Asher Wolinsky, and very different from the situation with com- a referee of this journal for the very useful comments. mon knowledge. 386 THE AMERICAN ECONOMIC RE VIEW JUNE 1989

The example is similar to the "coordi- cannot achieve an expected payoff hgher nated attack problem" which is well known than (1 - p)M. If the information could be- in the distributed systems literature.' A come common knowledge they would be able description of the problem and a compari- to achieve the payoff M. However, imagine son with ths paper analyzed appears in Sec- that the two players are located at two dif- tion IV. ferent sites and they communicate only by electronic mail signals. Due to "technical 1. Coordination Through Electronic Mail difficulties" there is a "small" probability E > 0,that the message does not arrive at its Two players, 1 and 2, are involved in a destination. At the risk of creating discord, coordination problem. Each has to choose the electronic mail network is set up to send between two actions A and B. There are two a confirmation automatically if any message possible states of nature, a and b. Each of is received, including not only the confirma- the states is associated with a payoff matrix tion of the initial message but a confirmation as follows: of the confirmation; and so on. To be more precise, it is assumed that, when player 1 gets the information that the state of nature The game G, is b, hs computer automatically sends a A B message (a blip) to player 2 and then player A M, M 0,- L 2's computer confirms the message and then B - L,O 090 player 1's computer confirms the confirmation and so on. If a message does not arrive, state a then the communication stops. No message probability 1- p is sent if the state of nature is a. At the end of the communication phase the screen dis- The game G, plays to the player the number of messages A B his machne has sent. Let T, be a variable A 0,o 0,- L for the number of messages i's computer B - L,O M, M sent (the number on i 's screen). Notice that sending the messages is not a state b strategic decision by the players. It is an probability p automatic device carried out by the comput- ers. The designer of the system sets up the co~municationnetwork between the players In the state of nature a(b) the players get and they can only choose between A and B a positive payoff, M, if both choose the after the communication phase has ended. action A(B). If they choose the same action If the two machnes exchange an infinite but it is the "wrong" one they get 0.If they number of messages, then we may say that fail to coordinate, then the player who played the two players have common knowledge B gets - L,where L > M. Thus, it is dan- that the game is G,. However, since only a gerous for a player to play B unless he is finite number of messages are transferred, confident enough that hs partner is going to the players never have common knowledge play B as well. The state a is the more likely that the game they play is G,. event: b appears with a priori probability of In choosing between A and B after the p <1/2. end of the communication phase, player 1 The information about the state of nature (and similarly player 2) faces uncertainty: is known initially only to player 1. Without given that he sent T, messages he does not transferring the information, the players know whether player 2 did not get the T,th message, or whether player 2 got the T,th message, but the T,th confirmation has been lost. Any number on the screen corresponds 'I should like to thank John Geanakopolos for refer- to a state of knowledge not only about the ring me to the "coordinated attack problem." state of nature but also about the other VOL. 79 NO. 3 RUBINSTEIN: ELECTRONIC MAIL GAME 387 player's knowledge. For example if player mail game is the set of natural numbers and 1's computer sent two messages it means the distribution of the pairs of types is de- that: duced from the electronic mail technology Kl(b)- 1 knows that b (namely, the probability of (TI, T2) being KlK2(b)- 1 knows that 2 knows that b respectively (0, O), (n + 1, n), and (n + 1, (by the fact that he has received confirma- n + 1) are 1- p, pe(1- E)~",and pe(1- tion of hs first message). However, it is not E)~"+',respectively). Define player i's strat- true that KlK2KlK2(b)- 1 does not know egy in the electronic mail game, S,, to be a that 2 knows that 1knows that 2 knows that function from the set of natural numbers b. Player 1 assigns probability z = E/[E+ O,1,2,. ..into the action space {A,B ). Then (1-&)&I to T2=l and (1-Z) to T2=2. S,(t) is interpreted as i's action if his ma- Therefore player 1 believes that: chine sent t messages. with probability 1- z K2K,K2(b) and with probability z that 11. The Analysis of the Electronic Mail Game 2 believes that with probability 1- z KlK2(b) and PROPOSITION 1: There is only one Nash with probability z that equilibrium in which player 1 plays A in the 1 believes that state of nature a. In this equilibrium the play- with probability z 2 believes that with prob- ers play A independently of the number of ability (1 - p)/(l - pe), a, and with proba- messages sent. bility (1 - z), 2 knows that b. The statements of hgher order are even PROOF: more complicated. Notice that, under the Let (S,, S2) be a Nash equilibrium such model's assumption that player 1 gets accu- that S,(O) = A. We will prove by induction rate information about the state of nature, that S,(t) = S2(t)= A for all t. If T2 = 0 "x" and "K,(x)" are two equivalent state- then player 2 did not get a message. He ments. knows that it might be because player 1 did Similarly, any number on a player's screen not send hm a message (ths could occur at the end of the communication stage corre- with probability 1- p) or because a message sponds to a sequence of propositions de- was sent but did not arrive (this happens scribing the player's knowledge about the with probability pe). In the first case, player state of nature, about hs opponent's belief 1 plays A (Sl(0) = A). If player 2 plays A, about the state of nature, about hs oppo- then, whatever Sl(l) is, player 2's expected nents's belief about hs belief about the op- payoff is at least; [(I - p)M + pe0]/[(1- p) ponent's belief about the state of nature and + pel and if he plays B he gets at most so on. The larger is TI, the more statements [- L(1- p)+ peM]/[(l- p)+ pel. There- of the type K,K2K,... KlK2(b) are true, fore it is strictly optimal for 2 to play A, that and the closer we are to the common knowl- is S2(0) = A. edge situation. Assume now that we have shown that, for How could we analyze the situation when all T,< t, players 1 and 2 play A in equilib- the two players have the numbers T, and T2 rium. Assume Tl = t. Player 1 is uncertain on their screens? To calculate hs best action whether T2 = t (in the case where player 2 when Tl = 2, for example, player 1may have received the t th message but 2's tth message to form beliefs about player 2's actions when was lost) or T2 = t -1 (in the case where 2 T2 is 1 or 2. The optimality of these would did not receive the tth message). Given that have to be checked given player 1's behavior he did not receive confirmation of hs tth when Tl =1, 2, or 3, and so on. Harsanyi's message, hs conditional probability that T2 method suggests that we analyze a situation =t -1 is z=e/[e+(l-e)e]>1/2. Thus it given any pair of numbers on the screens, as is more likely that player 1's last message did part of a game of incomplete information not arrive than that player 2 got the mes- whch I will refer to as "the electronic mail sage. (This fact is the key to our argument). game" (to distinguish from the coordination By the inductive assumption, player 1 as- games). The set of types in the electronic sesses that, if T2 = t - 1, player 2 will play A. 388 THE AMERICAN ECONOMIC REVIEW JUNE I989

If player 1 chooses B, player l's expected It is clear that if both divisions attack payoff is at most z(- L)+ (1- z)M. If he the enemy simultaneously they will win chooses A, then hs utility is 0. Given that a battle, whereas if only one division L > M and since z >1/2, his only best ac- attacks it will be defeated. The divi- tion must be A. Thus S,(t) = A. Similarly sions do not initially have plans for launchng an attack on the enemy, and we show that S,(t)= A. the commanding general of the first division wishes to coordinate a simul- Thus even if both players know that the taneous attack (at some time the next actual played coordination game is G, and day). Neither general will decide to even if the noise in the network (the proba- attack unless he is sure that the other bility E) is arbitrarily small, the players ig- will attack with hm. The generals can nore the information and play A. The best only communicate by means of a mes- expected payoff the players can obtain in senger. Normally, it takes the mes- any equilibrium is still (1 - p)M, just as if senger one hour to get from one en- no electronic mail system existed! campment to the other. However, it is vossible that he will get lost in the dark br, worst yet, be cavptured by the en- Remark 1: Consider the mechanism de- emy. Fortunately, on ths particular scribed above but with the addition that, night, everythng goes smoothly. How after a commonly known fixed finite number long it will take them to coordinate an of messages, T,the system stops, if it has not attack? stopped before. If E(- L) +(1- E)M > 0 Suppose the messenger sent by gen- then there is an equilibrium in whch each eral 1 makes it to general 2 with a player plays B if he receives confirmations message saying "Let's attack at dawn." of all hs messages. The expected payoffs of Will general 2 attack? Of course not, this equilibrium, conditional on the state b since general 1 does not know he got the message, and thus may not attack. are: (1 - E)=M to the last player who is sup- So general 2 sends the messenger back posed to get a message and (1- E)~-' with an acknowledgment. Suppose the [E(- L)+(1 - &)MIto the other player. messenger makes it. Will general 1 attack? No, because now general 2 does Notice that these two numbers are decreas- not know he got the message, so he ing in T and therefore the only "efficient" thinks general 1 may thnk that he schemes might be those with T =1and T = 2. (general 2) didn't get the original mes- The mechanism with T =1is a better scheme sage, and thus not attack. So general 1 for player 2 and T = 2 is a better scheme for sends the messenger back with an ac- player 1. If the communication channel is so knowledgment. But of course, ths is not enough either. I will leave it to the noisy that E(- L) (1- E)M < 0 then the + reader to convince himself that no efficient equilibrium is the one where the amount of acknowledgments sent back messages are ignored (the argument is simi- and forth ever guarantee agreement. lar to the proof of the proposition). Note that ths is true if the messenger succeeds in delivering the message ev- 111. The CoordinatedAttack Problem ery time.

As was mentioned in the introduction the electronic mail game is strongly related to The question asked in the quoted para- the coordinated attack problem known in graph is whether there is a common knowl- the distributed systems folklore. The prob- edge of the attack plan at the end of the lem as described in Joseph Halpern (1986, p. information transmission stage. The above 10) is the following: "communication protocol" cannot result in the players' having common knowledge about the time of the attack. However. the Two divisions of an army are camped fact that the generals could not acheve ;om- on two hlltops overlooking a common mon knowledge does not exclude the possi- valley. In the valley awaits the enemy. bility that with positive probability they will VOL. 79 NO. 3 RUBINSTEIN; ELECTRONIC MAIL GAME 389

both attack at dawn. Ths sounds plausible screens are " very large." Then a "very large" especially if the probability of a messenger number of statements of the type "player i failure is very small. knows that player j knows that.. .the coor- For ths reason it is interesting to analyze dination game is G," are correct. Still, the the problem in the explicit form of a game. players will not coordinate on the action B Ths is the minor contribution of ths paper. whereas they are able to coordinate on the In order to address the problem as a game, action B if it is common knowledge that the we need to add more structure to the prob- coordination game is G,. lem and, in particular, we have to specify the probability conditions under which general 1 B. The Electronic Mail Game as a decides to initiate an attack at dawn. In Perturbed Game terms of Section 11, state b can be interpreted as the conditions whch make an at- Selten's perfection definitions and the tack at dawn llkely to succeed, whle state a Kreps-Milgrom-Roberts-Wilson (1982) ap- is the "status quo" state. Action B is "at- proach used small perturbations in a game in tack at dawn" and action A is the default order to select an equilibrium in a game with action. The payoffs in Section I represent an multiplicity of equilibria and to create new assumption that, in case of an uncoordinated equilibria in the absence of a reasonable attack, only the general who attacks loses. If, equilibrium. If we think of E as being small alternatively, we assume that both generals' then the noisy electronic mail game is a utilities are - L if an uncoordinated attack perturbation of a non-noisy electronic mail is launched, then there is an equilibrium in game (the electronic mail game with E = 0). which general 2 attacks as soon as he gets at The non-noisy game has several equilibria least one message, provided that E is small (since it is just a coordination problem) how- enough (less than M/(M + L)). Ths last ever the ~erturbationunfortunatelv excludes fact emphasizes the importance of address- the more reasonable equilibria. Notice that ing the problem within a game-theoretic the difference between a game and a per- framework. turbed version of the game has already been demonstrated manv times in the ~astand I IV. Final Comments feel less paradoxicil about ths as'compared to the paradoxical features of the present A. Is "Almost Common Knowledge" example. Close to "Common Knowledge"? C. The Paradoxical Aspect of the Example It should be emphasized that the game about whch knowledge is being hypothe- What would you do if the number on sized in the above is the coordination game your screen is 17? It is hard to imagine that and not the electronic mail game. One is when L is slightly above M and E is small a concerned with what the two players do or player will not play B. The sharp contrast do not know about the payoffs in the coordi- between our intuition and the game-theo- nation game and with what the players do or retic analysis is what makes ths example do not know about the knowledge of their paradoxical. opponent. The story of the interchange of The example joins a long list of games messages by electronic mail is intended only such as the finitely repeated Prisoner's to provide a precise, albeit rather special, Dilemma, the chain store paradox, and model of how knowledge on those questions Rosenthal's game, in whch it seems that the may come to be shared by the players. source of the discrepancy is rooted in the The main message of ths paper is that fact that in our formal analysis we use math- players' strategic behavior under "almost ematical induction whle human beings do common knowledge" may be very different not use mathematical induction when rea- from that under common knowledge. To em- soning. Systematic explanation of our intu- phasize, by "almost common knowledge" I ition that we will play B when the number refer to the case when the numbers on the on our screen is 17 (ignoring the inductive 390 THEAMERICAN ECONOMIC REVIEW JUNE 1989 consideration contained within Proposition entire type space with the exception of 1's proof) is definitely a most intriguing (a,O,O) and is never common knowledge. question. Notice that when E = 0, the feasible states are just (a,O,O) and (b, a,a). D. Games with Incomplete Information F. Topology As mentioned earlier the situation without common knowledge is analyzed, A la Two of the readers of the first version of Harsanyi, as a game with incomplete in- ths paper, both experts in the literature on formation. Notice that almost all the common knowledge, raised objections to the non-abstract literature uses the distinction way I use the term "almost common knowl- between types to reflect differences in knowledge." They based their objection on the fact edge about payoff-relevant items. The that when E -+ 0 the information partitions current example is exceptional in that it of the players do not converge to the infor- demonstrates a family of natural game- mation partitions when E= 0 (see ths sec- theoretic scenarios in whch the main differ- tion, Part E). A referee suggested several ence between the types is in their knowledge topologes in which alternative concepts of about other players' knowledge. "almost common knowledge" make sense. Before reacting to this criticism let me E. A Formal Presentation of the Type emphasize again that I use the term "almost Spaces and the Information Partitions2 common knowledge" not for stating that the electronic mail game with E close to 0 is Those readers who are familiar with Au- almost the game with E= 0. What I am man (1976), may found it helpful to have a saying is that the situation with a high Tl is formal statement of the type spaces and the close to the common knowledge situation. information partitions in the electronic mail However, I would like to use this objection game. The type spaces of the two players are to spell out my opinion on the role that the sets which include (a, 0,O) and the triples topology (in common with most other fields (b, t, t') where t > 0 and t' is either t or of "fancy mathematics") should play in eco- t - 1. Array the set in the following order: nomic theory. Topology should be used in one of two ways: (1) as a techcal tool for phrasing a meta-claim about a family of models, or (2) as a substantial tool to formal- ize natural intuitions about "closeness." I envisage the hgh Tl situation as being close Player l's information partition is: to the common knowledge situation in the sense of (2). Ths may be unhelpful from a technical point of view and a conclusion from the example is indeed that the Nash equilibrium is not upper hemicontinuous in ths convergence. However, lack of technical and player 2's information partition is: usefulness is not an argument against the perception that a situation with high T, is close to a situation with common knowledge. Obviously other definitions of convergence may be useful not only as technical methods but also for expressing other intuitions of The meet of the two partitions is the trivial closeness. partition which contains only the entire type space. Thus the event "b" consists of the REFERENCES

21n this section I am closely following a referee's Aumann, Robert J., "Agreeing to Disagree," suggestion. Annals of Statistics, 1976, 4, 1236-239. VOL. 79 NO. 3 RUBINSTEIN: ELECTRONIC MAIL GAME 391

Binmore, Kenneth and Brandenberger, Adam, R., "Rational Cooperation in the Finitely "Common Knowledge and Game Theo- Repeated Prisoner's Dilemma," Journal of ry," Discussion Paper No. TE/88/167, Economic Theory, August 1982, 27, STICERD, London School of Economics, 245-52. 1987. Lewis, David, Convention, A Philosophical Halpern, Joseph Y., "Reasoning about Knowl- Study, Cambridge: Hanard University edge: An Overview," in Reasoning about Press, 1969. Knowledge, J. Y. Halpern, ed., Morgan Mertens, Jean-Francois and Zamir, Samuel, Kaufmann, 1986, 1-18. "Foundation of Bayesian Analysis for Harsanyi, J. C., "Games with Incomplete In- Games with Incomplete Information," In- formation Played by Bayesean Players," ternational Journal of Game Theory, 1985, Parts I, 11, 111, Management Science, 1967, 14, 1-29. 14, 159-82, 320-34, 486-502. Schiffer, Stephen R., Meaning, Oxford: Ox- Kreps, D., Milgrom, P., Roberts, J. and Wilson, ford University Press, 1972.

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