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Differential information and informational equilibrium
Economics Letters 1 (1978) 15-19 0 North-Holland Publishing Company
DIFFERENTIAL
INFORMATION
15
AND INFORMATIONAL
EQUILIBRIUM
H.M. SHEFRIN Department Received
of Economics,
University
ofRochester,
Rochester,
NY 14627,
UsA
May 1978
A concept of informational equilibrium is discussed based on the idea that each economic agent views himself as facing a Markovian decision problem. The case of differential information is considered. This definition is related to those found in general equilibrium theory and in the theory of stochastic games.
The existence of a fulfilled expectations equilibrium for an N-person stochastic game in which the players have differential information is discussed. Consider the extensive form of an N-person game. Partition the nodes of the game tree into time periods and let there be an infinite number bf periods. At the beginning of a period, nature selects an element x from a finite set X. Following this, player 1 chooses a decision dr from a finite set Dr ; then player 2 chooses d2, and so on until do is chosen at the end of the period in question. Nature then selects an element x’ from X to start the next period. Assume that x’ is chosen according to a probability distribution P{ . 1x, dl, d2, .... do} on X (where the conditioning is on the previous period’s decisions). The structure of the players’ information sets is crucial to the analysis. Let t be any period, and suppose that nature has selected the value of x for period t; however the players’ period t decisions have not yet been chosen. Clearly, the path followed in the game tree to this point specifies nature’s and all the players’ past decisions, as well as nature’s current choice of x. Let h denote the part of the path that specifies the players’ period t - 1 decisions, and nature’s choice of x in periods t - 1 and t. I assume that prior to taking his period t decision, player i receives an information signal pi where /3i is a function of h. Then player i’s period t information sets are related to pi in the following way. If nodes n and n’ are decision nodes for i in period t, and i receives the same information signal pi at n as he does at n’, then n and n’ both belong to the same information set. Therefore, player i may possess knowledge about period I - 1 decisions as well as the period t value of x; however, he has no knowledge of the other players’ period t decisions. Notice that the value of h is not specific to period t, but can occur in any period. Since pi is a function of h, some information signals will occur infinitely often, Also
16
H.M. Shefrin /Differential
information and informational’equilibrium
notice that pi is indexed by i; this permits two players to have different information during the same time period. Both of these features will turn out to be quite important. Suppose that in period t player i receives signal /3i and chooses decision die I assume that this combination results in a period t payoff (or utility) ui(pi, di). Therefore player i receives an infinite stream of payoffs. For purposes of exposition, let player i discount his period t payoff by the factor of-’ where 0 < cri < 1. My treatment of the individual player’s decision problem is significantly different from the standard approach. Specifically, I assume that every player acts as if he were involved in a Markov decision process. This means that subjectively he believes that he is playing against a ‘system’ whose behaviour is governed by a set of stationary conditional probability distributions. A typical distribution, for instance, takes the form Prob{ * I pi, di} where Prob {& I pi, di} denotes player i’s subjective probability that his next period’s information signal will be 0: given that his current signal is pi and he chooses current decision di. It is important to realize that i chooses his strategy as a best reply to this set of conditional distributions, and not as a best reply to the other players’ strategies. More precisely, suppose that i receives information signal pi during the first period. Let I’&) denote the expected utility that i receives when he chooses a best reply strategy to the above distributions. Then the function Vi( . ) satisfies the following functional equation:
for all pi* Let df be the maximizing decision for the above problem and define a policy function Fi by S&Ii) = d;. Even though pi was assumed to occur in the first period, it is well-known that it is also optimal for i to take decision d; in any period for which & occurs. Clearly, 6i is a strategy for i because it specifies a decision for i at each of his decision nodes in the game tree. In a Nash equilibrium each player knows the other players’ strategies, and chooses his own strategy as a best reply to theirs. In my framework, however, a player does not choose his strategy as a best reply to the others: he chooses his strategy as a best response to his subjective conditional distributions. Consequently, the Nash equilibrium is not an appropriate equilibrium concept for this problem. What then is an appropriate equilibrium concept for the problem? I would like to propose a type of fulfilled expectations or informational equilibrium. Recall that player i chooses his strategy on the basis of subjective probabilities like Prob (0; I pi, di} . Now it seems reasonable to say that i will maintain his beliefs and leave his strategy unaltered over time if he receives no disconfirming evidence about any of the probabilities Prob {/I: 1pi, di}. In particular, no such disconfirming evidence would come forth if the relative frequency with which 0: actually follows &
H.M. Shefrin /Differential
information and informational equilibrium
11
and di in the game tree coincides with i’s subjective probability Prob {fli I pi, di} for every pi, pi, and di. [This statement holds vacuously if the pair (j3, di) does not occur along the actual path in the tree.] In this case, player i is said to be in equilibrium. If all players are in equilibrium, the game is said to be in infkrzatiunal equilibrium because no player receives information which disconfirms his subjective beliefs. Proving that the game has an informational equilibrium is a fairly straightforward matter, and a sketch of the proof is easy to provide. First, notice that every assignment of subjective probability distributions to players results in the game tree having an associated finite Markov Chain. To see this, let h be given at the start of an arbitrary period. Then player i receives information signal /3i = &(h), and chooses decision di = Si(&). This completes the period, and nature begins the next period by selecting x’ according to the distribution P{ . I x, dl, d2, . . . . dN}. Clearly, this determines the next h = [x’, dI, d2, . . . . dN] . Hence the Markov Chain has h as a typical state with the transition probabilities generated by P. Since D1,Dz,....DN and X are all finite, the above Markov Chain is also finite; therefore, it has at least one closed set. Assume that the system settles in a closed set. Since every closed set possesses an invariant distribution, the relative frequency with which each signal /3: follows (pi, di) can be computed as a marginal distribution. This relative frequency can then be compared with player i’s subjective distribution Prob {& / pi, di}. If for all pi, pi, di, and i, the two agree [or the event (fli, di) does not occur in the closed set], then the game is in informational equilibrium. The remainder of the proof proceeds along the following lines. A test invariant distribution is selected and subjective distributions consistent with it are computed. The Markov Chain is then constructed and its closed sets are examined to see whether any objective invariant distribution that is so produced coincides with the test distribution. If there is such a distribution, the system is in equilibrium. Construct a correspondence in which the test distribution is mapped into the set of objective distributions derived in this manner. This correspondence turns out to be upper semi-continuous on a compact set (once mixed policies ~ that is, mixed strategies - are permitted). Since the correspondence maps distributions into distributions, it has a futed point. The fixed point clearly yields an informational equilibrium. The reason why I have developed the above framework is to study economies in which agents have differential information. Such economies have been the subject of considerable interest in recent years and it seems useful to draw a few comparisons. Radner (1968 and forthcoming) has studied the implications arising from the introduction of differential information about the state of nature into an ArrowDebreu model. [Also see Green (1977) and Kreps (1977).] One of the points Radner makes is that there are difficulties associated with endogenous changes in agents’information structures especially if there is present in the model a network structure in which agents exchange information. In contrast, no such difficulties arise in my model when networks are permitted.
18
H.M. Shefrin 1 Differential information and informational equilibrium
In a related vein, Hahn (1973) has proposed a notion of equilibrium which is better suited to the analysis of sequential ‘trading than is the traditional ArrowDebreu equilibrium. The model that I have developed captures the essential features of his discussion. Roughly, Hahn describes agents as being Bayesian dynamic programmers. Each agent receives messages from the other agents and nature. Every agent has his own ‘theory’ about the behaviour of the economy, and attempts to choose a subjective optimal policy relative to his ‘theory’. He also uses the messages he receives to ‘test’ his theory. If the messages do not disconfirm his theory, then the agent is said to be non-learning. Hahn indicates that by non-learning he means that observed frequencies and subjective probabilities agree. The system is in equilibrium when no agent is learning. Clearly, his non-learning equilibrium accords with my definition of informational equilibrium. My assumption that each player regards himself as being involved in a Markov decision process is rather different from the usual stochastic games approach [see Bewley and Kohlberg (1976)], however, it is extremely well-suited to the analysis of economic markets. For instance, consider current microeconomic theories about consumer and firm dynamic behaviour under uncertainty [Zabel(l972 and 1977)]. The consumer tends to regard price as a stable random variable (even though it is set by firms) and firms tend to regard quantity demanded as a stable random variable (even though it is determined by consumers). In these theories, neither the consumer nor firms consider the others’ strategies. Instead, policies are chosen as if the agent were involved in a Markov decision process. Clearly, one reason for this is that knowledge about the others’ strategies is simply unavailable: Information is incomplete [in Harsanyi’s (1967-68) sense]. The importance of my model is that it can bring both types of agents together in a market model and determine the subjective equilibrium probability distribution of each. I would like to close with two final points. First, market clearing at every instant of time is not a necessary condition for informational equilibrium. Indeed, market clearing may not even be possible because of the non-convex structure of the model (finiteness). This is a significant economic point because it allows agents to take into account the frequency with which they will not be able to carry out their desired plans. For example dynamic inventory and queuing models can be accomodated. Second, agents can themselves choose prices in my model; an appended auctioneer is not needed. Consequently, the framework is both flexible and robust.
References Bewley, T. and Elan Kohlberg, 1976, The asymptotic
theory of stochastic games, Mathematics of Operations Research 1, no. 3. Green, Jerry, 1977, The nonexistence of informational equilibria, Review of Economic Studies XLIV. Hahn, F.H., 1973, On the notion of equilibrium in economics - An inaugural lecture (Cambridge University Press, Cambridge).
H.M. Shefrin f Differential information and informational equilibrium Harsanyi, John C., 1967-1968, Games with incomplete information played by ‘Bayesian’ players, Management Science, Parts I-III, Management Science 14, 159-182, 320-334, and 4866502. Kreps, David, 1977, A note of ‘fulfilled expectations’ equilibria, Journal of Economic Theory 14. Radner, R., 1968, Competitive equilibrium under uncertainty, Econometrica 36, no. 1. Radner, R., forthcoming, Rational expectations equilibrium: Generic existence and the information revealed by prices, Econometrica. Shefrin, H.M., 1977, Games with self-generating distributions, mimeo. Zabel, E., 1972, Multiperiod monopoly under uncertainty, Journal of Economic Theory 5, no. 3. Zabel, E., 1977, Consumer behaviour under risk in disequilibrium trading, International Economic Review 18, no. 2.
19
DIFFERENTIAL
INFORMATION
15
AND INFORMATIONAL
EQUILIBRIUM
H.M. SHEFRIN Department Received
of Economics,
University
ofRochester,
Rochester,
NY 14627,
UsA
May 1978
A concept of informational equilibrium is discussed based on the idea that each economic agent views himself as facing a Markovian decision problem. The case of differential information is considered. This definition is related to those found in general equilibrium theory and in the theory of stochastic games.
The existence of a fulfilled expectations equilibrium for an N-person stochastic game in which the players have differential information is discussed. Consider the extensive form of an N-person game. Partition the nodes of the game tree into time periods and let there be an infinite number bf periods. At the beginning of a period, nature selects an element x from a finite set X. Following this, player 1 chooses a decision dr from a finite set Dr ; then player 2 chooses d2, and so on until do is chosen at the end of the period in question. Nature then selects an element x’ from X to start the next period. Assume that x’ is chosen according to a probability distribution P{ . 1x, dl, d2, .... do} on X (where the conditioning is on the previous period’s decisions). The structure of the players’ information sets is crucial to the analysis. Let t be any period, and suppose that nature has selected the value of x for period t; however the players’ period t decisions have not yet been chosen. Clearly, the path followed in the game tree to this point specifies nature’s and all the players’ past decisions, as well as nature’s current choice of x. Let h denote the part of the path that specifies the players’ period t - 1 decisions, and nature’s choice of x in periods t - 1 and t. I assume that prior to taking his period t decision, player i receives an information signal pi where /3i is a function of h. Then player i’s period t information sets are related to pi in the following way. If nodes n and n’ are decision nodes for i in period t, and i receives the same information signal pi at n as he does at n’, then n and n’ both belong to the same information set. Therefore, player i may possess knowledge about period I - 1 decisions as well as the period t value of x; however, he has no knowledge of the other players’ period t decisions. Notice that the value of h is not specific to period t, but can occur in any period. Since pi is a function of h, some information signals will occur infinitely often, Also
16
H.M. Shefrin /Differential
information and informational’equilibrium
notice that pi is indexed by i; this permits two players to have different information during the same time period. Both of these features will turn out to be quite important. Suppose that in period t player i receives signal /3i and chooses decision die I assume that this combination results in a period t payoff (or utility) ui(pi, di). Therefore player i receives an infinite stream of payoffs. For purposes of exposition, let player i discount his period t payoff by the factor of-’ where 0 < cri < 1. My treatment of the individual player’s decision problem is significantly different from the standard approach. Specifically, I assume that every player acts as if he were involved in a Markov decision process. This means that subjectively he believes that he is playing against a ‘system’ whose behaviour is governed by a set of stationary conditional probability distributions. A typical distribution, for instance, takes the form Prob{ * I pi, di} where Prob {& I pi, di} denotes player i’s subjective probability that his next period’s information signal will be 0: given that his current signal is pi and he chooses current decision di. It is important to realize that i chooses his strategy as a best reply to this set of conditional distributions, and not as a best reply to the other players’ strategies. More precisely, suppose that i receives information signal pi during the first period. Let I’&) denote the expected utility that i receives when he chooses a best reply strategy to the above distributions. Then the function Vi( . ) satisfies the following functional equation:
for all pi* Let df be the maximizing decision for the above problem and define a policy function Fi by S&Ii) = d;. Even though pi was assumed to occur in the first period, it is well-known that it is also optimal for i to take decision d; in any period for which & occurs. Clearly, 6i is a strategy for i because it specifies a decision for i at each of his decision nodes in the game tree. In a Nash equilibrium each player knows the other players’ strategies, and chooses his own strategy as a best reply to theirs. In my framework, however, a player does not choose his strategy as a best reply to the others: he chooses his strategy as a best response to his subjective conditional distributions. Consequently, the Nash equilibrium is not an appropriate equilibrium concept for this problem. What then is an appropriate equilibrium concept for the problem? I would like to propose a type of fulfilled expectations or informational equilibrium. Recall that player i chooses his strategy on the basis of subjective probabilities like Prob (0; I pi, di} . Now it seems reasonable to say that i will maintain his beliefs and leave his strategy unaltered over time if he receives no disconfirming evidence about any of the probabilities Prob {/I: 1pi, di}. In particular, no such disconfirming evidence would come forth if the relative frequency with which 0: actually follows &
H.M. Shefrin /Differential
information and informational equilibrium
11
and di in the game tree coincides with i’s subjective probability Prob {fli I pi, di} for every pi, pi, and di. [This statement holds vacuously if the pair (j3, di) does not occur along the actual path in the tree.] In this case, player i is said to be in equilibrium. If all players are in equilibrium, the game is said to be in infkrzatiunal equilibrium because no player receives information which disconfirms his subjective beliefs. Proving that the game has an informational equilibrium is a fairly straightforward matter, and a sketch of the proof is easy to provide. First, notice that every assignment of subjective probability distributions to players results in the game tree having an associated finite Markov Chain. To see this, let h be given at the start of an arbitrary period. Then player i receives information signal /3i = &(h), and chooses decision di = Si(&). This completes the period, and nature begins the next period by selecting x’ according to the distribution P{ . I x, dl, d2, . . . . dN}. Clearly, this determines the next h = [x’, dI, d2, . . . . dN] . Hence the Markov Chain has h as a typical state with the transition probabilities generated by P. Since D1,Dz,....DN and X are all finite, the above Markov Chain is also finite; therefore, it has at least one closed set. Assume that the system settles in a closed set. Since every closed set possesses an invariant distribution, the relative frequency with which each signal /3: follows (pi, di) can be computed as a marginal distribution. This relative frequency can then be compared with player i’s subjective distribution Prob {& / pi, di}. If for all pi, pi, di, and i, the two agree [or the event (fli, di) does not occur in the closed set], then the game is in informational equilibrium. The remainder of the proof proceeds along the following lines. A test invariant distribution is selected and subjective distributions consistent with it are computed. The Markov Chain is then constructed and its closed sets are examined to see whether any objective invariant distribution that is so produced coincides with the test distribution. If there is such a distribution, the system is in equilibrium. Construct a correspondence in which the test distribution is mapped into the set of objective distributions derived in this manner. This correspondence turns out to be upper semi-continuous on a compact set (once mixed policies ~ that is, mixed strategies - are permitted). Since the correspondence maps distributions into distributions, it has a futed point. The fixed point clearly yields an informational equilibrium. The reason why I have developed the above framework is to study economies in which agents have differential information. Such economies have been the subject of considerable interest in recent years and it seems useful to draw a few comparisons. Radner (1968 and forthcoming) has studied the implications arising from the introduction of differential information about the state of nature into an ArrowDebreu model. [Also see Green (1977) and Kreps (1977).] One of the points Radner makes is that there are difficulties associated with endogenous changes in agents’information structures especially if there is present in the model a network structure in which agents exchange information. In contrast, no such difficulties arise in my model when networks are permitted.
18
H.M. Shefrin 1 Differential information and informational equilibrium
In a related vein, Hahn (1973) has proposed a notion of equilibrium which is better suited to the analysis of sequential ‘trading than is the traditional ArrowDebreu equilibrium. The model that I have developed captures the essential features of his discussion. Roughly, Hahn describes agents as being Bayesian dynamic programmers. Each agent receives messages from the other agents and nature. Every agent has his own ‘theory’ about the behaviour of the economy, and attempts to choose a subjective optimal policy relative to his ‘theory’. He also uses the messages he receives to ‘test’ his theory. If the messages do not disconfirm his theory, then the agent is said to be non-learning. Hahn indicates that by non-learning he means that observed frequencies and subjective probabilities agree. The system is in equilibrium when no agent is learning. Clearly, his non-learning equilibrium accords with my definition of informational equilibrium. My assumption that each player regards himself as being involved in a Markov decision process is rather different from the usual stochastic games approach [see Bewley and Kohlberg (1976)], however, it is extremely well-suited to the analysis of economic markets. For instance, consider current microeconomic theories about consumer and firm dynamic behaviour under uncertainty [Zabel(l972 and 1977)]. The consumer tends to regard price as a stable random variable (even though it is set by firms) and firms tend to regard quantity demanded as a stable random variable (even though it is determined by consumers). In these theories, neither the consumer nor firms consider the others’ strategies. Instead, policies are chosen as if the agent were involved in a Markov decision process. Clearly, one reason for this is that knowledge about the others’ strategies is simply unavailable: Information is incomplete [in Harsanyi’s (1967-68) sense]. The importance of my model is that it can bring both types of agents together in a market model and determine the subjective equilibrium probability distribution of each. I would like to close with two final points. First, market clearing at every instant of time is not a necessary condition for informational equilibrium. Indeed, market clearing may not even be possible because of the non-convex structure of the model (finiteness). This is a significant economic point because it allows agents to take into account the frequency with which they will not be able to carry out their desired plans. For example dynamic inventory and queuing models can be accomodated. Second, agents can themselves choose prices in my model; an appended auctioneer is not needed. Consequently, the framework is both flexible and robust.
References Bewley, T. and Elan Kohlberg, 1976, The asymptotic
theory of stochastic games, Mathematics of Operations Research 1, no. 3. Green, Jerry, 1977, The nonexistence of informational equilibria, Review of Economic Studies XLIV. Hahn, F.H., 1973, On the notion of equilibrium in economics - An inaugural lecture (Cambridge University Press, Cambridge).
H.M. Shefrin f Differential information and informational equilibrium Harsanyi, John C., 1967-1968, Games with incomplete information played by ‘Bayesian’ players, Management Science, Parts I-III, Management Science 14, 159-182, 320-334, and 4866502. Kreps, David, 1977, A note of ‘fulfilled expectations’ equilibria, Journal of Economic Theory 14. Radner, R., 1968, Competitive equilibrium under uncertainty, Econometrica 36, no. 1. Radner, R., forthcoming, Rational expectations equilibrium: Generic existence and the information revealed by prices, Econometrica. Shefrin, H.M., 1977, Games with self-generating distributions, mimeo. Zabel, E., 1972, Multiperiod monopoly under uncertainty, Journal of Economic Theory 5, no. 3. Zabel, E., 1977, Consumer behaviour under risk in disequilibrium trading, International Economic Review 18, no. 2.
19