Information, efficiency and the core of an economy: Comments on Wilson s paper

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1 Information, efficiency and the core of an economy: Comments on Wilson s paper Dionysius Glycopantis 1 and Nicholas C. Yannelis 2 1 Department of Economics, City University, Northampton Square, London EC1V 0HB, UK ( d.glycopantis@city.ac.uk) 2 Department of Economics, University of Illinois at Urbana-Champaign, IL 61820, USA ( nyanneli@uiuc.edu) 1 Introduction In his seminal paper, Wilson (1982) discusses the issues of exchange efficiency and the core in the context of differential information economies. His work has attracted a lot of attention and had widespread influence in the development of economic theory. In this note we offer some explanations and make comments on the Wilson s paper which we believe will help to clarify futher his ideas. In particular we discuss two of his examples which he uses to make a number of points centered around feasibility, efficiency and his coarse core. We assume that the reader is acquainted with this paper. Examples and tables are numbered as they appear there. The main point is that Wilson s analysis and use of the term rational expectations equilibria (REE) are not the same as in the Radner-Allen approach. 2 Explanations on Example 2 and 3 In Example 2 we are explaining Wilson s calculations of prices and quantities in a market of state-contigent claims. We also point out that when Wilson uses the term rational expectations equilibria (REE) his definition is not the same as REE given by Radner (1979) and Allen (1981), and used in Glycopantis - Muir - Yannelis in this volume. It is precisely the differenial information of the agents which accounts for the variety of possible equilibrium ideas. In Example 3 we give a detailed analysis, confirming Wilson s results, through the interpretation of the game in a tree form. The use of extensive form games lends itself naturely in situations where one of the players sends or does not send a signal to the other one. We wish to thank A. Muir and R. Wilson for their very helpful comments and suggestions.

2 2 D. Glycopantis and N.C. Yannelis Example ( 2. We explain here how Wilson obtains the prices (p a,p b,p c ) = 1, , 115) 25 under state s = a. Notice that this is not rational expectations equilibria and we will return to this point below. Wilson says that state a has been chosen by nature. P1 sees a, P2 observes {a, c} and P3 the event {a, b}. However each player will also try to sell his endowments in the remaining states because he knows that they are valuable to somebody else. The calculations below show how to obtain the prices p a =1, p b = p c = The agents receive price signals from the auctioneer and maximixe their interim expected utility subject to their budget constraints. We have in effect the following problems. P1: Maximize u 1 = logx 1a p a x 1a =5p a +1p b +3p c P2: Maximize u 2 = logx 2a + logx 2c p a x 2a + p c x 2c =3p a +5p b +1p c P3: Maximize u 3 = logx 3a + logx 3b p a x 3a + p b x 3b = p a +3p b +5p c The agents send back to the Walrasian auctioneer their quantities demanded as signals. The equilibrium conditions are: For the quantities, in State a: 5pa+1p b+3p c p a + 3pa+5p b+1p c 2p a + pa+3p b+5p c 2p a =9, =9and in State c: 3pa+5p b+1p c 2p c =9. in State b: pa+3p b+5p c 2p b These relations are satisfied by p a =1,p b = ,p c = and the implied allocation of claims is x 1a = ,x 2a = ,x 3a = ,x 1b = x 2b =0,x 3b = 9, x 1c = x 3c =0, x 2c =9. We have obtained in these calculations the prices above and the Allocations of Claims in Table III. The first column in the Market Allocation corresponds to the first column of Allocation of claims. It says that when it is revealed that state a has been realized, then the first column of Allocation of Claims is what is relevant. Given the endowments in Table II we can do per prevailing state the calculations and arrive eventually at columns b and c of Market Allocation in Table III. The above analysis is not in the area of rational expectations in the usual sense. First the endowments in Wilson s formulation are not private information measurable and also, which is probably more significant, we now have a function

3 Information, efficiency and the core of an economy: Comments on Wilson s paper 3 p : Ω IR l per prevailing state. In REE there is only one price function defined on Ω. Hence, although REE is itself an interim concept, here we have an alternative interim concept. Suppose we were to define a REE in the context of Wilson. It could go as follows. We are looking for p : Ω IR l such that each agent, given the element of his information set which he observes and the signal that he receives from prices, maximizes his interim expected utility subject to his relevant budget constraint and such that when the state is revealed the markets clear. Now we would not insist on any kind of measurability of the allocations. According to the above definition (p a,p b,p c )=(1, 1, 1) is a non-revealing REE set of prices and the Allocation in Table II is the corresponding REE quantities. The agents rely only on their information sets and maximize interim expected utility subject to their budget constraints without insisting on measurability of their choices. This is consistent with the fact that the endowments are not measurable. With respect to the statement in Footnote 3, what is meant there is that given prices p =(1, 1, 1), the agents are only maximizing interim expected utility, when the state of nature is uncertain, and otherwise they keep their endowment. The outcome is again the Allocation in Table II. Perhaps defining a new REE notion is more satisfactory than prohibiting traders, under some circumstances from trading. Below Table III, Wilson refers to REE but in a different sense to the one described above. This follows from the nature of the prices he proposes. The vector p =(1, 0, 0) clears the market only under the condition s = a. Indeed it can be replaced by any non-negative price vector p =(1,k 1,k 2 ) with k i 1and analogous prices identifying the other states. Everybody demands his own endowment. Also, if all prices are positive and different, and again ignoring the lack of measurability of the initial allocation, we have a fully revealing REE, and these initial endowments are confirmed as an equilibrium. Example 3. This example calculates a Nash equilibrium in the context of a normal form game in which players have differential information. Originally the player of the columns is not allowed to send a signal and in the second instance he can signal to the player of the rows. The payoffs in Table VI are in a single commodity. Agents 1 and 2 have stricty concave and increasing utility functions u 1 and u 2 on this good. As there is no confusion, the payoffs in the trees below are given in terms of the commodity. In order to make the analysis clearer, we cast it in a tree form attaching to Nature, as the third agent, the possility to choose in beginning between states left (L) and right (R) with equal probabilities. We call P2 the Column player and P1 the Row player. The information sets are given by F 1 = {{L, R}} and F 2 = {{L}, {R}}. P1 cannot distinguish between L and R that nature chose, but P2 can do so. Case 1. P2 makes no announcement. Payoffs are determined from nature and the decision of P1 who, given the probabilities of choices, maximizes his expected utility. The Nash equilibrium, indicated in Figure 1, is for P1 to play d. P2 is completely passive and does nothing.

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5 Information, efficiency and the core of an economy: Comments on Wilson s paper 5 á â ã ä å æ é ð ë ì î é ê ë ì î ç è ç è é ð ë é ñ ò ó ñ ô é ê ë é ö ø é ð ë é ö ø é ê ë é ñ ò ó ñ ô é ù ú ù é û ú ù é ù ú ù é û ú ù Figure 3 Case 2. P2 is allowed to send a signal to P1. We now construct Figure 2. Payoffs are determined from nature and the decision of P1. He cannot distinguish between L and R that nature chose but can hear the announcement of P2 who can either tell the truth or choose to lie. We consider various possibilities of pure strategies, where the first choice of each player refers to his first information set, from left to right: P2, (L, R) (u, d) for P 1, not Nash; P2, (L, L) (d, d) for P 1, Nash; P2, (L, L) (d, u) for P1, not Nash; P2, (R, L) (u, d) for P1, not Nash; P2, (R, R) (d, d) for P1, Nash; P2, (R, R) (u, d) for P1, not Nash. One of the possible Nash equilibria is indicated on Figure 2 with heavy lines. It can be obtained by folding up the tree to the one in Figure 3. The above confirms the statement of Wilson that in this case also P1 will play d. There are other Nash equilibria as well. In these P1 will always play d but P2 can use mixed strategies as well. Furthermore all these Nash equilibria, with appropriate probabilities (beliefs) attached to the nodes of the information sets, are also perfect Bayesian equilibria (PBE). In the same example Wilson says: In the coarse core of the corresponding cooperative game (an equilibrium) is the strategy which chooses Up or Down as

6 6 D. Glycopantis and N.C. Yannelis ý þ ÿ Figure 4 the state is Left or Right, and which gives to Row an insured payoff of 3 units in either case;.... An interpretation is as follows. 1 Cooperation allows us to discard the payoff vectors (0, 0) and (0, 2), and e L 1 =2, e R 1 =4, e L 2 =2and e R 2 =0can be thought of as the players s endowments in the single commodity. The agents act on the basis the meet of the information algebras, F 1 = {{L, R}}. The corresponding tree is shown in Figure 4. The players know the probabilities and have to reach a decision concerning both nodes. The commodity payoff contraints are 0 c L 1 + c L 2 4 and 0 d R 1 + d R 2 4. The expected utilities of the playes corresponding to their random endowments are E 1 = 1 2 u 1(2) u 1(4), and E 2 = 1 2 u 2(2) u 2(0) The coarse core allocations are obtained as solutions to the problem 1 We recall that Wilson defines the coarse core to consist of allocations which cannot be blocked by any coalition of agents who act on the basis of the intersection of their algebras of information sets. Wilson imposes no measurability conditions on allocations.

7 Information, efficiency and the core of an economy: Comments on Wilson s paper 7 Maximize E 1 = 1 2 u 1(u)+ 1 2 u 1(d) 1 2 u 2(u)+ 1 2 u 2(d) E 2 (fixed) E u 1(2) u 1(4) and E u 2(2) u 2(0). Then there is a coarse core allocation in which P1 gets (3, 3) and P2 the allocation (1, 1). We can see this as follows. We maximize the expected utility of one agent subject to a given expected value for the other. A particular solution is the one above. We note that Example 1, which we did not discuss, illustrates the problem of adverse selection, and requires no special interpretation. References Allen, B.: Generic existence of completely revealing equilibria with uncertainty, when prices convey information. Econometrica 49, (1981) Glycopantis, D., Yannelis, N.C.: Equilibrium concepts in differential information economies. This volume Radner, R.: Rational expectation equilibrium: generic existence and information revealed by prices. Econometrica 47, (1979) Wilson, R.: Information, Efficiency, and the core of an economy. Econometrica 46, (1978)