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