A Note on the E-mail Game
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believe的意思Bounded Rationality and Induction
Uwe Dulleck¤y
Comments welcome
Abstract
In Rubinstein´s(1989)E-mail game there exists no Nash equilib-rium where players u strategies that condition on the E-mail com-
munication.In this paper I restrict the utilizable information for one
player.I show that in contrast to Rubinstein´s result,in a payo¤
birth是什么意思dominant Nash equilibrium players u strategies that condition on
the number of messages nt.Therefore-induction under the as-
sumption of bounded rational behavior of at least one player leads to
happy meals
a more intuitive equilibrium in the E-mail game.
Keywords:Induction,Subgame Perfect Equilibrium,Information ts,Imperfect recall
JEL Classi cation:C72
海鲜英语of Economic Theory,Spandauer Str.1,D-10178Berlin,Germany,Ph.:+49-30-20935657,Fax.:+49-30-20935619,e-mail: dulleck@wiwi.hu-berlin.de
y I am grateful for helpful comments by Jörg Oechssler,Ulrich Kamecke,Elmar Wolf-stetter and minar participants at University College London.Financial support by the Deutsche Forschungsgellschaft(DFG)through SFB373is gratefully acknowledged.
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1Introduction
In his Electronic Mail game Rubinstein (1989) illustrates the di¤erence be-tween common knowledge
and almost common knowledge .Using his ex-ample I illustrate another puzzling e¤ect on the equilibrium behavior of this game by applying a notion of imperfect recall to the model.I show that bounded rational behavior in this game almost reestablishes the equilibrium that exists under common knowledge and full rationality.In the Electronic Mail game two players either play a game G a (with probability (1¡p )>12or G b (with probability p <12In each game players choo between action A and B .In both games it is mutually bene cial for players to choo the same action.Figure 1describes the game.In game a (b )the Pareto dominant equilibrium is the one where players coordinate on A (B ).If players cho di¤erent actions the player who played B is punished by ¡L regardless of the game played.The other player gets 0.It is assumed that the potential loss L is not less than the gain M and both are positive.
Figure 1:The Email Game
Only player 1is informed about the game that is actually played.After the state of the world is determined two machines (one for each player)com-municate about the game.If game b prevails,player 1´s machine nds an contains a textbook prentation of the problem.
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E-mail message(a beep)to player2´s machine which is automatically con- rmed.This con rmation is con rmed and so on.With a small probability "a message gets lost.Communication stops,when one of the messages(the original message or one of the con rmations)is lost.Players are informed how many messages their machine nt to the other player.Then they have to make their decision.
mbaaThe Electronic Mail game reprents a slight deviation from common knowledge( almost common knowledge in Rubinstein´s terms).Combined with perfect rationality this leads to discontinuous drop in expected payo¤s. Paradoxically in this ca the game has an equilibrium,where players never play the payo¤dominant equilibrium in one game(b)even if many messages were nt.
The point I make is that by reducing the ability to process information the existence of an additional subgame perfect equilibrium is guaranteed.The extension I propo is that a player cannot distinguish among the elements in a certain t of he cannot distinguish wether T;T+1;:::;T+l messages were nt.If a su¢cient number of messages is nt,players in this new equilibrium coordinate on the payo¤dominant equilibrium in both games and therefore that equilibrium Pareto dominates an equilibrium where players do not play the payo¤dominant equilibrium.
As in related work by Dulleck and Oechssler(1996)the E-mail game is an example where induction under bounded rationality leads to di¤erent results. Therefore the hypothesis implied by experimental data that agents do not u induction correctly,may be due to the fact that they face limitations on utilizable information which are due to bounded rationality.The E-mail game shows that agents might u induction correctly but in a di¤erent en-vironment.
One further result follows from the main results of the paper:
Given the following descending order of the quality of the informational structure: common knowledge , almost common knowledge , almost com-mon knowledge and non-distinguishability ,and no knowledge at all the expected payo¤of the equilibrium under the di¤erent regimes vary non-monotonically.This is in contrast to results prented in the economic lit-erature where either knowing less about a characteristic of the state of the world is an advantage but then knowing even less usually does not worn the McKelvey and Palfrey(1992)and Ronthal(1981)among others prent experiments on the centipede game that imply this hypothesis.
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outcome for a player.Or in other cas,knowing more is better but usually knowing even more does not worn the result.Note the additional reduc-tion I propo is in the same dimension as the reduction in Rubinstein´s original contribution.
The propod argument can also be applied to solve the related paradox of the Coordinated Attack problem(in Fagin et al(1995),Chapter 6).
2The Electronic Mail Game and its exten-sion
Using the notation of Rubinstein(1989)the feasible states s of the world are reprented as a triple consisting of the game actually played and the number of messages nt by player1and by s2f(a;0;0);(b;1;0);(b;1;1); (b;2;1);(b;2;2);:::(b;T1;T2):::g.T1and T2are the numbers obrved by player1and player2respectively,T22f T1¡1;T1g.For simplicity of no-tation I will only u a pair consisting of the numbers of messages nt.We must be in game b if and only if T1¸1.Hereby we rule out that the machine of player1fails to nd a message although we are in state b.Note however that we do not rule out that this message gets lost.淘宝培训
Figure2gives a graphical reprentation of this game,where the auto-matic moves(by nature)of the machines are reprented.The outcomes are the numbers players obrve before making their decisions.
wampFigure2:Information Outcomes of the email game
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In Rubinstein´s game,player1cannot distinguish between the out-comes (T1;T1¡1)and(T1;T1)(and player2cannot distinguish between the states(T2;T2)and(T2+1;T2)).In this ca he always only obrves T1 (T2).A player choos his strategy conditioned on the number of messages nt by his machine.A strategy will be played in two states of the world-the two states were the player i obrves that T i messages were nt by his machine.He has to build beliefs about which informational outcome is the actual one.
The extend the E-mail game by adding non-distinguishability of num-bers.In the extended tup one player is not able to distinguish the numbers t2f T;T+1;:::;T+l g where l2N.This information structure is com-mon knowledge.I refer to this version as the extended game.Otherwi the players play the game as it is described above.
The non-distinguishability reprents the ca where a player cannot ob-rve or interpret the information about the number of messages nt if they belong to the interval[T;T+l].This might be due to the fact that he is not able to distinguish the numbers(interpret the numbers in the right way)or
that the machine is not able to show di¤erent symbols if t is in the critical interval.
This modi cation ems to be obvious given l!1,which is the ca where the machine or the player lo track at stage T.Justi cations for this assumption could be the over ow of the machine´s capacities(it can only count up to a certain number)or that real players actually stop counting after they nt a certain number of messages.The result in this ca is identical to Rubinstein´s(1989)problem where the maximum number of messages to be nt is limited.I show that the weaker condition that players cannot distinguish between some states is enough to yield a subgame perfect equilibrium with coordination.This weaker condition may be due to minor problems in the processing of a player can only obrve even numbers.
Language di¤erences may be a reason why a player cannot distinguish between,let us say,he maybe unsure of the right order of17 and18)!.Or the machine may not be able to show18and therefore it stays on17for two turns and then jumps to19."
!Assume that one plays the Email game in China using traditional chine numbers (which were taught before)-I am sure one would get confud interpreting the symbols.
"The propod logic can also be applied to a situation where one or both players ly even
numbers.Necessary for the prent results is that the information ts
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Given this modi cation of non-distinguishability one has a problem which analysis is similar to that of the problem of imperfect recall #in the n that a player forgets how many beeps he has heard or messaged he received before but he is reminded once in a while about the actual number.The player cannot distinguish/remember whether his machine nt T;T +1;:::or T +l messages and therefore he has to choo one action for all obrvations in the interval.
3Results
In the original game,Rubinstein (1989)proves that there is no Nash equi-librium where players condition on the number of messages nt:
Proposition 1(Rubinstein (1989))There is only one Nash equilibrium in which player 1plays A in game G a .In this equilibrium players play A independently of the number of messages nt.
五年级家长会班主任发言稿The formal proof is provided in Rubinstein (1989).The basic idea of the proof is that in states (0;0)and (1;0)the obvious equilibrium is (A;A )-given p <12.Using this as the start of an induction,one
has that up to the obrvation of T ¡1for each player it is optimal to play A:The consistent belief z =""+"(1¡")to be at the rst of two indistinguishable outcomes (T;T ¡1)and (T;T )for player 1[or (T ¡1;T ¡1)and (T;T ¡1)for player 2]is greater than 12.Given the stated belief and that up to state (T ¡1;T ¡1)[or (T;T ¡1)for player 2]the best reply of the other player is A ,it is a best a answer to choo A if the information t is reached becau this decision is independent of the strategy of the other player at the cond indistinguishable outcome in the information t.By induction this is true for every obrved T .
information ts of the other player in a way that the rst part (the states that are in the corresponding rst information t of the other player)is smaller than the rest of the information t.Therefore the next informational structure that would yield the Rubinstein result is where both player cannot distinguish three succeeding numbers and the information ts overlap exactly the way that in each t three outcomes are in each of the corresponding ts of the other player.#Piccione and Rubinstein (1996)and Aumann et al.(1996)in addition to a special issue of Games and Economic Behavior 1996(forthcoming)cover the problem of imperfect recall in an example of an abnt-minded driver.An application to the centipede game can be found in Dulleck and Oechssler (1996).nt
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