Eco514-Game Theory Lecture 5: Games with Payoff Uncertainty(2 Marciano siniscalchi September 30, 1999 Introduction This lecture continues our analysis of games with payoff uncertainty. The three main objec tives are: (1)to illustrate the flexibility of the Harsanyi framework(or our version thereof) (2) to highlight the assumptions implicit in the conventional usage of the framework, and the possible departures; 3) to discuss its potential problems, as well as some solutions to the latter Cournot revisited Recall our Cournot model with payoff uncertainty. Firm 2's cost is known to be zero; Firm I s cost is uncertain, and will be denoted by cE, 23. Demand is given by P(Q)=2-Q and each firm can produce qi E 0, 1 We represent the situation as a game with payoff uncertainty as follows: let 32=10, 1 Ti=0J, 213, T2=(Q2) and P2(0)=T. It is easy to see that specifying P1 is not relevant for the purposes of Bayesian Nash equilibrium analysis: what matters there are the beliefs conditional on each ti E T1, but these will obviously be degenerate The following equalities define a Bayesian Nash equilibrium(do you see where these com oIn q1(0) q1(G) 92 1 q1(0)+(1-丌)q1(
Eco514—Game Theory Lecture 5: Games with Payoff Uncertainty (2) Marciano Siniscalchi September 30, 1999 Introduction This lecture continues our analysis of games with payoff uncertainty. The three main objectives are: (1) to illustrate the flexibility of the Harsanyi framework (or our version thereof); (2) to highlight the assumptions implicit in the conventional usage of the framework, and the possible departures; (3) to discuss its potential problems, as well as some solutions to the latter. Cournot Revisited Recall our Cournot model with payoff uncertainty. Firm 2’s cost is known to be zero; Firm 1’s cost is uncertain, and will be denoted by c ∈ {0, 1 2 }. Demand is given by P(Q) = 2 − Q and each firm can produce qi ∈ [0, 1]. We represent the situation as a game with payoff uncertainty as follows: let Ω = {0, 1 2 }, T1 = {{0}, { 1 2 }}, T2 = {Ω} and p2(0) = π. It is easy to see that specifying p1 is not relevant for the purposes of Bayesian Nash equilibrium analysis: what matters there are the beliefs conditional on each t1 ∈ T1, but these will obviously be degenerate. The following equalities define a Bayesian Nash equilibrium (do you see where these come from?): q1(0) = 1 − 1 2 q2 q1( 1 2 ) = 3 4 − 1 2 q2 q2 = 1 − 1 2 πq1(0) + (1 − π)q1( 1 2 ) 1
For T=5, we get (0)=0625;91()=0.375;=075 (by comparison, a is the equilibrium quantity for both firms if Firm 1's cost is always c=0.) Textbook analysis Recall that, for any probability measure q E A(Q)and player i E N, we defined the event lai= w: pi(wlti w))=q. In this game, [p2]2=Q: that is, at any state of the world, Player 2's beliefs are given by p2. By way of comparison, it is easy to see that it cannot be the case that Ipili=@(why? regardless of how we specify pi For notational ease(and also as a "sneak preview"of our forthcoming treatment of interactive epistemology), we introduce the belief operator. Recall that, for every iE N and wES, ti(@)denotes the cell of the partition Ti containing w Definition 1 Given a game with payoff uncertainty G=(N, Q2, (Ai, ui, Ti)ieN), Player i's belief operator is the map Bi: 22-2 defined by VECO, B(E)=wEQ: Pi(Elt;(w))=1] A more appropriate name for Bi( would perhaps be certainty operator, but we shall follow traditional usage. Ifw E B (E), we shall say that "At w, Player i is certain that(or believes that)E is true. "Certainty is thus taken to denote probability one belief Now return to the Cournot game and take the point of view of Player 1. Since Ip2]2=Q2 it is trivially true that B1(]2)=Q; in words, at any state w E Q, Player 1 is certain that Player 2's beliefs are given by p2(i.e. by T ). By the exact same argument, at any state w, Player 2 is certain that Player 1 is certain that Player 2s beliefs are given by p2: that is, B2(B1(P22)=Q2). And so on and so forth The key point is that Harsanyi's model of games with payoff uncertainty, together with a specification o f the players'priors, easily generates infinite hierarchies of interactive beliefs that is "beliefs about beliefs edy Although you may not immediately"see"these hierarchies, they are there-and they are ilv retrieved uncertainty concerning Firm I's payoffs, but no uncertainty about Firm 2s beliee s there is We can summarize the situation as follows: in the setup under consideration
For π = 1 2 , we get q1(0) = 0.625; q1( 1 2 ) = 0.375; q2 = 0.75 (by comparison, 2 3 is the equilibrium quantity for both firms if Firm 1’s cost is always c = 0.) Textbook analysis Recall that, for any probability measure q ∈ ∆(Ω) and player i ∈ N, we defined the event [q]i = {ω : pi(ω|ti(ω)) = q}. In this game, [p2]2 = Ω: that is, at any state of the world, Player 2’s beliefs are given by p2. By way of comparison, it is easy to see that it cannot be the case that [p1]1 = Ω (why?) regardless of how we specify p1. For notational ease (and also as a “sneak preview” of our forthcoming treatment of interactive epistemology), we introduce the belief operator. Recall that, for every i ∈ N and ω ∈ Ω, ti(Ω) denotes the cell of the partition Ti containing ω. Definition 1 Given a game with payoff uncertainty G = (N, Ω,(Ai , ui , Ti)i∈N ), Player i’s belief operator is the map Bi : 2Ω → 2 Ω defined by ∀E ⊂ Ω, Bi(E) = {ω ∈ Ω : pi(E|ti(ω)) = 1} A more appropriate name for Bi(·) would perhaps be certainty operator, but we shall follow traditional usage. If ω ∈ Bi(E), we shall say that “At ω, Player i is certain that (or believes that) E is true.” Certainty is thus taken to denote probability one belief. Now return to the Cournot game and take the point of view of Player 1. Since [p2]2 = Ω, it is trivially true that B1([p2]2) = Ω; in words, at any state ω ∈ Ω, Player 1 is certain that Player 2’s beliefs are given by p2 (i.e. by π). By the exact same argument, at any state ω, Player 2 is certain that Player 1 is certain that Player 2’s beliefs are given by p2: that is, B2(B1([p2]2) = Ω). And so on and so forth. The key point is that Harsanyi’s model of games with payoff uncertainty, together with a specification of the players’ priors, easily generates infinite hierarchies of interactive beliefs, that is “beliefs about beliefs...” Although you may not immediately “see” these hierarchies, they are there—and they are easily retrieved. We can summarize the situation as follows: in the setup under consideration, there is uncertainty concerning Firm 1’s payoffs, but no uncertainty about Firm 2’s beliefs. 2
There is also uncertainty about Firm I's beliefs-but in a degenerate sense: there is a one-one relationship between Player 1s conditional beliefs at any w Q and her cost At first blush, this makes sense: after all, payoff uncertainty is about Player 1s cost, so as soon as Firm 1 learns the value of c, her uncertainty is resolved. Similarly, since Firm 2s cost is known to be zero, there is no payoff uncertainty as far as the latter is concerned. How about the absence of uncertainty about Firm 2s beliefs? This is a legitimate as sumption, of course. The point is, it is only an assumption: it is not a necessary consequence of rationality, of the Bayesian approach, or, indeed, a necessary feature of Harsanyi's model of incomplete information "Unconventional""(but legit)use of Harsanyi's approach Indeed, it is very easy to enrich the model to allow for uncertainty (on Firm 1's part)about Firm 2s beliefs Let us consider the following alternative model for our Cournot game. First, Q=wary E 10, 5:a,y E(1, 2. The interpretation is that in state wary, Firm 1's cost is c,Firm 1’s" belief state”isr, and firm2s“ belief state”isgy. This terminology is nonstandard and merely suggestive: the exact meaning will be clear momentarily Next,letn1={{ual,uc}:c∈{0,},x∈{1,2}}={:c∈{0,是},x∈{1,2}and I learns her cost and bewley]: yE11, 2))=(t2: y E(1, 2). Thus, at each state w, Firm T2={t er“ belief state”, and firn2 learns his“ belief state We can get a lot of action from this simple extension. Let us define conditional proba- bilities as follow P2(uon|+2)=p2(u21+)=0.5 i. e. type t) of Firm 2 is certain that Firm 1 is in belief state I whenever her cost is 0,in belief state 2 whenever her cost is 2; moreover, the two combinations of belief states and costs are equally likely. Next p2(ai212)=1-p2(22)=075 i.e. t2 is certain that Firm 1s belief state, regardless of cost, is 3= 2; moreover, he has a different marginal on c than type t). Finally n(cn()=p1(a14)=1andp1a)=ph(un142)=1 that is, regardless of her cost, in belief state 1 Firm 1 is certain that she is facing type t? hereas in belief state 2 she considers both types of Firm 2 to be equally likely. As I noted last time, this is really not relevant(also see the "Common priors"section below) To complete the specification of our priors, we simply assume that players regard the cells of their respective type partitions as being equally likely
There is also uncertainty about Firm 1’s beliefs—but in a degenerate sense: there is a one-one relationship between Player 1’s conditional beliefs at any ω ∈ Ω and her cost. At first blush, this makes sense: after all, payoff uncertainty is about Player 1’s cost, so as soon as Firm 1 learns the value of c, her uncertainty is resolved. Similarly, since Firm 2’s cost is known to be zero, there is no payoff uncertainty as far as the latter is concerned. How about the absence of uncertainty about Firm 2’s beliefs? This is a legitimate assumption, of course. The point is, it is only an assumption: it is not a necessary consequence of rationality, of the Bayesian approach, or, indeed, a necessary feature of Harsanyi’s model of incomplete information. “Unconventional” (but legit) use of Harsanyi’s approach Indeed, it is very easy to enrich the model to allow for uncertainty (on Firm 1’s part) about Firm 2’s beliefs. Let us consider the following alternative model for our Cournot game. First, Ω = {ωcxy : c ∈ {0, 1 2 }; x, y ∈ {1, 2}}. The interpretation is that in state ωcxy, Firm 1’s cost is c, Firm 1’s “belief state” is x, and Firm 2’s “belief state” is y. This terminology is nonstandard and merely suggestive: the exact meaning will be clear momentarily. Next, let T1 = {{ωcx1, ωcx2} : c ∈ {0, 1 2 }, x ∈ {1, 2}} = {t cx 1 : c ∈ {0, 1 2 }, x ∈ {1, 2}} and T2 = {{ω01y, ω02y, ω 1 2 1y , ω 1 2 2y} : y ∈ {1, 2}} = {t y 2 : y ∈ {1, 2}}. Thus, at each state ω, Firm 1 learns her cost and her “belief state”, and Firm 2 learns his “belief state.” We can get a lot of action from this simple extension. Let us define conditional probabilities as follows: p2(ω011|t 1 2 ) = p2(ω 1 2 21|t 1 2 ) = 0.5 i.e. type t 1 2 of Firm 2 is certain that Firm 1 is in belief state 1 whenever her cost is 0, in belief state 2 whenever her cost is 1 2 ; moreover, the two combinations of belief states and costs are equally likely. Next, p2(ω022|t 2 2 ) = 1 − p2(ω 1 2 22|t 2 2 ) = 0.75 i.e. t 2 2 is certain that Firm 1’s belief state, regardless of cost, is x = 2; moreover, he has a different marginal on c than type t 1 2 . Finally, p1(ω011|t 01 1 ) = p1(ω 1 2 11|t 1 2 1 1 ) = 1 and p1(ω021|t 02 1 ) = p1(ω 1 2 21|t 1 2 2 1 ) = 1 2 that is, regardless of her cost, in belief state 1 Firm 1 is certain that she is facing type t 1 2 , whereas in belief state 2 she considers both types of Firm 2 to be equally likely. As I noted last time, this is really not relevant (also see the “Common priors” section below). To complete the specification of our priors, we simply assume that players regard the cells of their respective type partitions as being equally likely. 3
The following equalities define a BNE 1(1) g2(t) 1 n(+2)+g2(t 31 42 t2) 692 292 t2 q(t2)+7q( You should be able to see where the above equalities come from by inspecting the defi- nitions of pi and p2 With the help of a numerical linear equation package we get (1)=.62638 q1(2)=63472 37638 q1(2 .38472 Note that we have simply applied Or's definition of Bayesian Nash equilibrium. That is we are still on familiar ground. We have only deviated from "tradition"in that our model is more elaborated than the "textbook variant Consider state woll. Observe that won E B1(t2): that is, in this state Firm 1 is certain that Firm 2's marginal on c is 3-2, and indeed this belief is correct. Moreover, Firm 2 is certain that, if Firm 1 has low cost, she(Firm 1)holds correct beliefs about his(Firm 2s) marginal on c, this belief, too, is correct. However, Firm 2 thinks that, if Firm 1 has high cost, she(Firm 1)may be mistaken about his(Firm 2 s) marginal on c with probability Thus, there seem to be"minimal "deviations from the textbook treatment given above; in particular, Firm 2 s first-order beliefs about c are the same in both cases. Yet, the equilibrium outcome in state woll is different from the"textbook" prediction. Indeed, there is no state in which Bayesian Nash equilibrium predicts the same outcome as in the " textbook"treatment
The following equalities define a BNE: q1(t 01 1 ) = 1 − 1 2 q2(t 1 2 ) q1(t 02 1 ) = 1 − 1 2 1 2 q2(t 1 2 ) + 1 2 q2(t 2 2 ) q1(t 1 2 1 1 ) = 3 4 − 1 2 q2(t 1 2 ) q1(t 1 2 2 1 ) = 3 4 − 1 2 1 2 q2(t 1 2 ) + 1 2 q2(t 2 2 ) q2(t 1 2 ) = 1 − 1 2 1 2 q1(t 01 1 ) + 1 2 q1(t 1 2 2 1 ) q2(t 2 2 ) = 1 − 1 2 3 4 q1(t 02 1 ) + 1 4 q1(t 1 2 2 1 ) You should be able to see where the above equalities come from by inspecting the defi- nitions of p1 and p2. With the help of a numerical linear equation package we get q1(t 01 1 ) = .62638 q1(t 02 1 ) = .63472 q1(t 1 2 1 1 ) = .37638 q1(t 1 2 2 1 ) = .38472 q2(t 1 2 ) = .7472 q2(t 2 2 ) = .7138 Note that we have simply applied OR’s definition of Bayesian Nash equilibrium. That is, we are still on familiar ground. We have only deviated from “tradition” in that our model is more elaborated than the “textbook” variant. Consider state ω011. Observe that ω011 ∈ B1(t 1 2 ): that is, in this state Firm 1 is certain that Firm 2’s marginal on c is 1 2 − 1 2 , and indeed this belief is correct. Moreover, Firm 2 is certain that, if Firm 1 has low cost, she (Firm 1) holds correct beliefs about his (Firm 2’s) marginal on c; this belief, too, is correct. However, Firm 2 thinks that, if Firm 1 has high cost, she (Firm 1) may be mistaken about his (Firm 2’s) marginal on c with probability 1 2 . Thus, there seem to be “minimal” deviations from the textbook treatment given above; in particular, Firm 2’s first-order beliefs about c are the same in both cases. Yet, the equilibrium outcome in state ω011 is different from the “textbook” prediction. Indeed, there is no state in which Bayesian Nash equilibrium predicts the same outcome as in the “textbook” treatment. 4
The bottom line is that (i) assumptions about higher-order beliefs do influence equilib- rium outcomes, and (ii) it is very easy to analyze deviations from the"textbook"assumptions about higher-order beliefs in the framework of standard Bayesian Nash equilibrium analysis Priors and common priors The other buzzword that is often heard in connection with games with incomplete informa- tion is“ common prior Simply stated, this is the assumption that pi= p for all i E N. Note that, strictly peaking, the common prior assumption(CPA for short)is part of the"textbook"definition of a "game with incomplete information our slightly nonstandard terminology emphasizes that(1) we do not wish to treat priors(common or private)as part of the description of the model, but rather as part of the solution concept; and that(2) we certainly do not wish to impose the CPa in all circumstances. But is the CPa at all reasonable? The answer is, well, it depends. One often hears the following argument Prior beliefs reflect prior information. We assume that players approach the game with a common heritage, i. e. with the same prior information. Therefore, their prior beliefs should be the same On priors as summary of previously acquired information Let us first take this argument at face value. From a Bayesian standpoint, this makes sense only if we assume that players approach the game after having observed the same events for a very long time; only in this case, in fact, will their beliefs converge But perhaps we do not want to be really Bayesians; perhaps "prior information"is"no information"and we wish to invoke some variant of the principle of insufficient reason. I personally do not find this argument all that convincing, but you may differ On Priors and interactive beliefs However, the real problem with this sort of justification of the CPA is that, as we have illustrated above, the set Q actually conveys information about both payoff uncertainty and the players' infinite hierarchies of interactive beliefs. Therefore, it is not clear how players beliefs about infinite hierarchies of beliefs can "converge due to a long period of commor observations. " How do I"learn"your beliefs? The bottom line is that the only way to assess the validity of the CPa is via its imp ions for the infinite hierarchies of beliefs it generates IMore precisely: perhaps I can make inferences about your beliefs, with the aid of some auxiliary as ptions, but I can never observe your beliefs
The bottom line is that (i) assumptions about higher-order beliefs do influence equilibrium outcomes, and (ii) it is very easy to analyze deviations from the “textbook” assumptions about higher-order beliefs in the framework of standard Bayesian Nash equilibrium analysis. Priors and Common Priors The other buzzword that is often heard in connection with games with incomplete information is “common prior.” Simply stated, this is the assumption that pi = p for all i ∈ N. Note that, strictly speaking, the common prior assumption (CPA for short) is part of the “textbook” definition of a “game with incomplete information”; our slightly nonstandard terminology emphasizes that (1) we do not wish to treat priors (common or private) as part of the description of the model, but rather as part of the solution concept; and that (2) we certainly do not wish to impose the CPA in all circumstances. But is the CPA at all reasonable? The answer is, well, it depends. One often hears the following argument: Prior beliefs reflect prior information. We assume that players approach the game with a common heritage, i.e. with the same prior information. Therefore, their prior beliefs should be the same. On priors as summary of previously acquired information Let us first take this argument at face value. From a Bayesian standpoint, this makes sense only if we assume that players approach the game after having observed the same events for a very long time; only in this case, in fact, will their beliefs converge. But perhaps we do not want to be really Bayesians; perhaps “prior information” is “no information” and we wish to invoke some variant of the principle of insufficient reason. I personally do not find this argument all that convincing, but you may differ. On Priors and interactive beliefs However, the real problem with this sort of justification of the CPA is that, as we have illustrated above, the set Ω actually conveys information about both payoff uncertainty and the players’ infinite hierarchies of interactive beliefs. Therefore, it is not clear how players’ beliefs about infinite hierarchies of beliefs can “converge due to a long period of common observations.” How do I “learn” your beliefs?1 The bottom line is that the only way to assess the validity of the CPA is via its implications for the infinite hierarchies of beliefs it generates. 1More precisely: perhaps I can make inferences about your beliefs, with the aid of some auxiliary assumptions, but I can never observe your beliefs! 5
We now know that the CPa is equivalent to the assumption that, at any state w EQ, players are not willing to engage in bets over the realization of w-contingent random variable Thus, it entails a very strong notion of agreement For instance, in our elaboration of the Cournot example, denote by pt the prior proba- bility of cell ti of Ti, and by q the prior probability of cell t2 of T2. Then the conditional probabilities indicated above imply that the priors pi and p2 must satisfy the following State p1()p2(a Po P 0 2P02 q W012 0 0 (1-q) P2H(1-q)3 Table 1: The priors in the Elaborated Cournot Model Disregarding the last column for the time being, the table yields a set of necessary conditions for the existence of a common prior. First, from the line corresponding to w121, P12 Hence, from the line corresponding to w122, we must have 5q=4(1-q), or However, from the line corresponding to wo22, Po2=23=1. This is impossible, because the per's must add up to one. Thus, there can be no common prior The rightmost column of table 1 illustrates a bet that in accordance with the results cited above, at each state w yields a positive expected payoff to Player 1 conditional on t1(w) and a negative expected payoff to Player 2 conditional on t2(w). Hence, at each state w of the model, Player 1 would want to offer a bet stating that, at each state w, Player 2 would pay her X(w) dollars, and Player 2 would readily accept it To see this, note that E[]=1, E[Xt]=1, E(X+02]=5 and E(Xt]=5 the other hand, E[X t2=-1 and E[Xt2=- If players' conditional beliefs were consistent with a common prior, no such mutually acceptable bet would exist. This statement is also known as the no-trade theorem. Indeed the results cited above state that the converse is also true: if no such mutually acceptable bet exists, then players' conditional beliefs are consistent with a common prior. For instance, in the simpler"textbook"treatment of the Cournot model (in which the set of states is the set
We now know that the CPA is equivalent to the assumption that, at any state ω ∈ Ω, players are not willing to engage in bets over the realization of ω-contingent random variables. Thus, it entails a very strong notion of agreement. For instance, in our elaboration of the Cournot example, denote by p cx the prior probability of cell t cx 1 of T1, and by q the prior probability of cell t 1 2 of T2. Then the conditional probabilities indicated above imply that the priors p1 and p2 must satisfy the following equalities: State ω p1(ω) p2(ω) X(ω) ω011 p01 1 2 q 1 ω 1 2 11 p 1 2 1 0 1 ω021 1 2 p02 0 3 ω 1 2 21 1 2 p 1 2 2 1 2 q -2 ω012 0 0 0 ω 1 2 12 0 0 0 ω022 1 2 p02 3 4 (1 − q) -2 ω 1 2 22 1 2 p 1 2 2 1 4 (1 − q) 3 Table 1: The priors in the Elaborated Cournot Model Disregarding the last column for the time being, the table yields a set of necessary conditions for the existence of a common prior. First, from the line corresponding to ω 1 2 21, p 1 2 2 = q. Hence, from the line corresponding to ω 1 2 22, we must have 1 2 q = 1 4 (1 − q), or 2q = 1 − q, so q = 1 3 = p 1 2 2 . However, from the line corresponding to ω022, p02 = 3 2 2 3 = 1. This is impossible, because the pcx’s must add up to one. Thus, there can be no common prior. The rightmost column of Table 1 illustrates a bet that, in accordance with the results cited above, at each state ω yields a positive expected payoff to Player 1 conditional on t1(ω), and a negative expected payoff to Player 2 conditional on t2(ω). Hence, at each state ω of the model, Player 1 would want to offer a bet stating that, at each state ω 0 , Player 2 would pay her X(ω 0 ) dollars, and Player 2 would readily accept it. To see this, note that E[X|t 01 1 ] = 1, E[X|t 1 2 1 1 ] = 1, E[X|t 02 1 ] = 1 2 and E[X|t 1 2 2 1 ] = 1 2 ; on the other hand, E[X|t 1 2 ] = −1 and E[X|t 2 2 ] = − 3 4 . If players’ conditional beliefs were consistent with a common prior, no such mutually acceptable bet would exist. This statement is also known as the no-trade theorem. Indeed, the results cited above state that the converse is also true: if no such mutually acceptable bet exists, then players’ conditional beliefs are consistent with a common prior. For instance, in the simpler “textbook” treatment of the Cournot model (in which the set of states is the set 6
of possible costs for Firm 1), for a bet X: 10, 51-R to be profitable to Firm 1 conditional on every ti(w)=w it must be the case that X(O)>0 and X()>0; but then no such bet can be acceptable to Firm 2 Justifying Harsanyi Models There are two related problems with the models of payoff uncertainty we have proposed The first has to do with generality. We have seen that we can construct rather involved models of interactive beliefs in the presence of payoff uncertainty. In particular, our models generate complicated hierarchies of beliefs rather easily However, let us ask the reverse question. Given some collection of hierarchies of beliefs can we always exhibit a model which generates in in some state? The second question has to do with one implicit assumption of the model. In order to make statements about Player 1's beliefs concerning Player 2s beliefs, Player 1 must be assumed to"know"p2 and T2. In order to make statements about Player 1s beliefs about Player 2's beliefs about Player 1's beliefs, we must assume that Player 2"knows"pi and Ti and that Player 1"knows"this More concisely: the model itself must be "common knowledge". But we cannot even formulate this assumption Both issues are addressed in a brilliant paper by Mertens and Zamir(Int. J. Game Theory, 1985). The basic idea is as follows Let us focus on two-player games. First, let us fix a collection Q2 of payoff parameters This set is meant to capture "physical uncertainty, or more generally any kind of uncertainty that is not related to players' beliefs a player's beliefs about Q2 are by definition represented by points in A(Q0).Now, here's the idea: if we wish to represent a player's beliefs about: (1)Q2 and(2) her opponent's beliefs about oo it is natural to consider the set g2=90×△(9°) and describe a player's beliefs about(1)and(2)above as points in A(@). The idea readily generalizes: suppose we have constructed a set S as above: then and we represent beliefs about(1)S and(2)the opponent's beliefs about 2 as points in A(Q+). Of course these are complicated spaces(even if Q20 is finite), but in any case the definitions are straightforward We wish to describe a player's beliefs by an infinite sequence e=(p, p2,...)of probability measures such that, for each k>1, PE 4(Q2k-). That is, each player's beliefs are described
of possible costs for Firm 1), for a bet X : {0, 1 2 } → R to be profitable to Firm 1 conditional on every t1(ω) = ω it must be the case that X(0) > 0 and X( 1 2 ) > 0; but then no such bet can be acceptable to Firm 2. Justifying Harsanyi Models There are two related problems with the models of payoff uncertainty we have proposed. The first has to do with generality. We have seen that we can construct rather involved models of interactive beliefs in the presence of payoff uncertainty. In particular, our models generate complicated hierarchies of beliefs rather easily. However, let us ask the reverse question. Given some collection of hierarchies of beliefs, can we always exhibit a model which generates in in some state? The second question has to do with one implicit assumption of the model. In order to make statements about Player 1’s beliefs concerning Player 2’s beliefs, Player 1 must be assumed to “know” p2 and T2. In order to make statements about Player 1’s beliefs about Player 2’s beliefs about Player 1’s beliefs, we must assume that Player 2 “knows” p1 and T1, and that Player 1 “knows” this. More concisely: the model itself must be “common knowledge”. But we cannot even formulate this assumption! Both issues are addressed in a brilliant paper by Mertens and Zamir (Int. J. Game Theory, 1985). The basic idea is as follows. Let us focus on two-player games. First, let us fix a collection Ω 0 of payoff parameters. This set is meant to capture “physical uncertainty,” or more generally any kind of uncertainty that is not related to players’ beliefs. A player’s beliefs about Ω0 are by definition represented by points in ∆(Ω0 ). Now, here’s the idea: if we wish to represent a player’s beliefs about: (1) Ω 0 and (2) her opponent’s beliefs about Ω 0 , it is natural to consider the set Ω 1 = Ω0 × ∆(Ω0 ) and describe a player’s beliefs about (1) and (2) above as points in ∆(Ω1 ). The idea readily generalizes: suppose we have constructed a set Ω k as above: then Ω k+1 = Ωk × ∆(Ωk ) and we represent beliefs about (1) Ω k and (2) the opponent’s beliefs about Ω k as points in ∆(Ωk+1). Of course these are complicated spaces (even if Ω 0 is finite), but in any case the definitions are straightforward. We wish to describe a player’s beliefs by an infinite sequence e = (p 1 , p2 , . . .) of probability measures such that, for each k ≥ 1, p k ∈ ∆(Ωk−1 ). That is, each player’s beliefs are described 7
by a measure on S, a measure on99×△(g9), a measure on9"×△(9)x△(920×△(99) and so on. Let Eu= llk>o a(s)denote the set of such descriptions: the reason for the superscript will be clear momentarily You may immediately see a problem with this representation of beliefs. Fix e=(p,p2,.)E E. Observe that, for every k≥2,p∈△(2-1)=△(92k-2×△(92k-2).Thus, the marginal of p on @2k-2 conveys information about a player's beliefs about @2k-2. However, by definition so does p-1∈△(9k-2 We obviously want the information conveyed by p and p- about our player's beliefs concerning 9-2 to be consistent. We thus concentrate on the subset of Eo defined by E marggk-2P =p Mertens and Zamir prove that, under regularity conditions on Q, the set E is homeo morphic to the set A(Q20 x E): that is, there exists a one-to-one, onto function f: E- △(9°×E0)( continuous, and with a continuous inverse) such that(1) every sequence e∈E corresponds to a unique measure f(e) on the product space Q20 x EO, and(2) every point Moreover, as you would expect, if e=(,p ,.), the to a unique e=f-w,e)EEl (u°,c0)=(u°,p2,yp2…)∈g∈E° can be mapped back marggk-1f(e)=p (recall the definition of Eo) In plain English, this says that, if we are interested in describing a player's beliefs about (1)Q and (2) her opponents full hierarchy of beliefs about S, then we should look no further than E. That is, we can regard elements as E as our player's types But. there is still something wrong with the construction so far. The reason is that hile each sequence e EE satisfies the consistency requirement, its elements may assign positive probability to inconsistent(sub)sequences of measures. That is, a player may hold consistent beliefs, but may believe that her opponent does not On the other hand, it is easy to see that, in any state w of a model of payoff-uncertainty as we have defined it, players' beliefs are consistent. Hence, at any state, players necessarily believe that their opponents hold consistent beliefs, that their opponent believe that their beliefs are consistent, and so on. That is, there is common certainty of consistency. We wish to impose the same restriction on the sequences of probability measures we are considering It is easy to do so inductively. Assume we have defined Ek.Then E+1={e∈E1:f(e)(9×E)=1} This makes sense: f(e)is a measure on 20 x E, so we can use it to read off the probability of the event Q20 x E. Note well what we are doing: the restriction is stated in terms of the probability measure on Q20 x EU induced by eE E, but(via the function f) this entails a restriction on the elements of the sequence e=(p, p,...)
by a measure on Ω 0 , a measure on Ω 0 × ∆(Ω0 ), a measure on Ω0 × ∆(Ω0 ) × ∆(Ω0 × ∆(Ω0 )), and so on. Let E 0 = Q k≥0 ∆(Ωk ) denote the set of such descriptions: the reason for the superscript will be clear momentarily. You may immediately see a problem with this representation of beliefs. Fix e = (p 1 , p2 , . . .) ∈ E. Observe that, for every k ≥ 2, p k ∈ ∆(Ωk−1 ) = ∆(Ωk−2×∆(Ωk−2 )). Thus, the marginal of p k on Ωk−2 conveys information about a player’s beliefs about Ω k−2 . However, by definition, so does p k−1 ∈ ∆(Ωk−2 )! We obviously want the information conveyed by p k and p k−1 about our player’s beliefs concerning Ω k−2 to be consistent. We thus concentrate on the subset of E 0 defined by E 1 = {e ∈ E 0 : ∀k ≥ 2, margΩk−2 p k = p k−1 } Mertens and Zamir prove that, under regularity conditions on Ω, the set E 1 is homeomorphic to the set ∆(Ω0 × E 0 ): that is, there exists a one-to-one, onto function f : E 1 → ∆(Ω0 ×E 0 ) (continuous, and with a continuous inverse) such that (1) every sequence e ∈ E 1 corresponds to a unique measure f(e) on the product space Ω 0 × E 0 , and (2) every point (ω 0 , e0 ) = (ω 0 , p1 , p2 ...) ∈ Ω 0 ∈ E 0 can be mapped back to a unique e = f −1 (ω 0 , e0 ) ∈ E 1 . Moreover, as you would expect, if e = (p 1 , p2 , . . .), then margΩk−1 f(e) = p k (recall the definition of E 0 ). In plain English, this says that, if we are interested in describing a player’s beliefs about (1) Ω 0 and (2) her opponent’s full hierarchy of beliefs about Ω0 , then we should look no further than E 1 . That is, we can regard elements as E 1 as our player’s types. But, there is still something wrong with the construction so far. The reason is that, while each sequence e ∈ E 1 satisfies the consistency requirement, its elements may assign positive probability to inconsistent (sub)sequences of measures. That is, a player may hold consistent beliefs, but may believe that her opponent does not. On the other hand, it is easy to see that, in any state ω of a model of payoff-uncertainty as we have defined it, players’ beliefs are consistent. Hence, at any state, players necessarily believe that their opponents hold consistent beliefs, that their opponent believe that their beliefs are consistent, and so on. That is, there is common certainty of consistency. We wish to impose the same restriction on the sequences of probability measures we are considering. It is easy to do so inductively. Assume we have defined E k . Then E k+1 = {e ∈ E 1 : f(e)(Ω0 × E k ) = 1} This makes sense: f(e) is a measure on Ω0 × E 0 , so we can use it to read off the probability of the event Ω 0 × E k . Note well what we are doing: the restriction is stated in terms of the probability measure on Ω 0 × E 0 induced by e ∈ E 1 , but (via the function f) this entails a restriction on the elements of the sequence e = (p 1 , p2 , . . .). 8
Finally, let E=nk Ek(how do we know the intersection is nonempty? ) The e main esult is as follows Theorem 0.1(Mertens and Zamir, 1985 Under regularity conditions, there exists a home- amorphism g:E→△(90×E) such that, for all k1ande=(p2,p2,)∈E, marg gk-1g(e)=p This is really what we were after. Define &= SxexE(we treat the first E as referring to Player 1's hierarchical beliefs, and the second as referring to Player 2's). Next, for all e1, e t(u)={u′=(如0,21,2):1=e} for each i= 1, 2; that is, in state w, Player i"learns"only her own beliefs. Finally, let T={=(0,1,l2):E=e}:e∈E} i.e. the set of possible types corresponds to E. This is just a big huge model of payoff uncertainty, but it is a model according to our definition. The key points are (1) Every possible" reasonable"hierarchical belief is represented in this big huge (2)Type partitions arise naturally: they correspond to "reasonable"hierarchical Thus, both difficulties with models of payoff uncertainty can be overcome
Finally, let E = T k≥1 E k (how do we know the intersection is nonempty?). The main result is as follows: Theorem 0.1 (Mertens and Zamir, 1985). Under regularity conditions, there exists a homeomorphism g : E → ∆(Ω0 × E) such that, for all k ≥ 1 and e = (p 1 , p2 , . . .) ∈ E, marg Ωk−1 g(e) = p k This is really what we were after. Define Ω = Ω0×E×E (we treat the first E as referring to Player 1’s hierarchical beliefs, and the second as referring to Player 2’s). Next, for all ω = (ω 0 , e1, e2), let ti(ω) = {ω 0 = (¯ω 0 , e¯1, e¯2) : ¯ei = ei} for each i = 1, 2; that is, in state ω, Player i “learns” only her own beliefs. Finally, let Ti = {{ω 0 = (¯ω 0 , e¯1, e¯2) : ¯ei = ei} : ei ∈ E} i.e. the set of possible types corresponds to E. This is just a big huge model of payoff uncertainty, but it is a model according to our definition. The key points are: (1) Every possible “reasonable” hierarchical belief is represented in this big huge model. (2) Type partitions arise naturally: they correspond to “reasonable” hierarchical beliefs. Thus, both difficulties with models of payoff uncertainty can be overcome. 9
