Game Theory Without Partitions, and Applications to Speculation and Consensus

1 Errors in Information Processing

There are a number of errors that are typically made by decision makers that suggest that we go beyond the orthodox Bayesian paradigm. We list some of them:

1. Agents ignore the subtle information content of some signals, and perceive only their face value. For example, an order to “produce 100 widgets” might convey all kinds of information about the mood of the boss, the profitability of the widget industry, the health of fellow workers and so on, if the receiver of the message has the time and capacity to think about it long enough. Another important example involves prices. It is very easy to compute the cost of a basket of goods at the going prices, but it takes much longer to deduce what the weather must have been like all across the globe to explain those prices. In Bayesian decision making, it is impossible to perform the first calculation without also performing the second.

2. Agents often do not notice when nothing happens. For example, it might be that there are only two states of nature: either the ozone layer is disintegrating or it is not. One can easily imagine a scenario in which a decaying ozone layer would emit gamma rays. Scientists, surprised by the new gamma rays would investigate their cause, and deduce that the ozone was disintegrating. If there were no gamma rays, scientists would not notice their absence, since they might never have thought to look for them, and so might incorrectly be in doubt as to the condition of the ozone.

3. What one knows is partly a matter of choice. For example, some people are notorious for ignoring unpleasant information. Often there are other psychological blocks to processing information.

4. People often forget.

5. Knowledge derived from proofs is not Bayesian. A proposition might be true or false. If an agent finds a proof for it, he knows it is true. But if he does not find a proof, he does not know it is false.

6. People cannot even imagine some states of the world.

We can model some aspect of all of these non-Bayesian methods of information processing by generalizing the notion of partition from the usual Bayesian analysis. Let Ω , a finite set, represent the set of all possible (physical) states of the world. Let P :   Ω → 2 Ω / { Ø } be an arbitrary “possibility correspondence,” representing the information processing capacity of an agent. For each ω ∈ Ω , P ( ω ) is interpreted to mean the collection of states the agent thinks are possible when the true state is ω. Let P ¯ denote the range of P, so P ¯ = { R ⊂ Ω | ∃ ω ∈ Ω ,   R = P ( ω ) } . Given an arbitrary event A ⊂ Ω , we say that the agent knows A at ω if P ( ω ) ⊂ A , since for any ω ′ ∈ Ω which he regards as possible at ω, ω ′ ∈ A .

Consider, for example, Ω = { a ,   b } as the state space. Let the possibility correspondence P :   Ω → 2 Ω take P ( a ) = { a } and P ( b ) = { a ,   b } . We can interpret ω = a to mean the ozone layer is disintegrating, or a horse is winning, or a proposition is true. Similarly, we can interpret ω = b to mean the ozone is not disintegrating, or the horse is losing, or the theorem is false.

Since P ( a ) = { a } , when ω = a the agent knows that his horse is winning, or that the ozone is disintegrating, or that the theorem is true. But when ω = b the agent has no idea whether his horse is winning or losing, or what is happening to the ozone, or whether the theorem is true or false. The reason for the asymmetry in the agent's information processing could be interpreted as any one of the above categories of errors. The agent might take notice of the gamma rays in state a, but not notice that there were no gamma rays in state b. In the horse racing interpretation of the model, the agent might not be able to face the unpleasant news that his favorite horse is damaged. Or he might not remember an event where nothing of interest happened to him.

In Bayesian decision theory, the information possibility correspondence always defines a partition: for every ω, ω ′ ∈ Ω , ω ∈ P ( ω ) and either P ( ω ) ∩ P ( ω ′ ) = Ø or else P ( ω ) = P ( ω ′ ) . A Bayesian decision-maker could not have the possibility correspondence in the above example. He would reason at ω = b that since he did not receive the signal { a } , that in fact the state must be b. In Bayesian decision theory the information at a state ω is always consistent with what could be deduced from knowing the model and the signal:

For an arbitrary possibility correspondence, such as in the above example, this need not be the case.

Observe that in the generalized possibility approach to knowledge, there need not be any presumption that the agent understands the entire state space Ω . It may well be that the sets of possibilities in P ¯ are all confined to some small subset of Ω . In that case there would be ω ∈ Ω such that ω ∉ P ( ω ′ ) , for any ω ′ ∈ Ω . Such ω are not even imaginable by the agent. Similarly if ω ∉ P ( ω ) , then when ω actually occurs the agent does not think of it, although he might at other times.

In the next few sections we shall describe how decision theory and game theory can be extended to generalized partitions. Many phenomenon (such as betting) which cannot be observed in equilibrium when every agent has partition information will now become possible. To give content to our extension, however, it is necessary to categorize precisely the kinds of information processing errors which can occur, and which kinds of errors permit each new phenomenon. Betting, for example, can be an equilibrium even when agents always imagine the truth, provided they make other errors. On the other hand betting is ruled out by a degree of rationality that falls short of partition information.

We shall now describe three limitations on the possibility correspondence. Later we shall introduce two more.

Definition We say that P is nondeluded if ω ∈ P ( ω ) for all ω ∈ Ω . Under this hypothesis the agent who processes information according to P always considers the true state as possible.

Definition (Knowing that you know (KTYK): When Knowledge is Self-Evident) If for all ω ∈ Ω , and all ω ′ ∈ P ( ω ) , we have P ( ω ′ ) ⊆ P ( ω ) , then we say that the agent knows what he knows. If the agent knows some A at ω, and can imagine ω ′ , then he would know A at ω ′ . Bacharach (1985), Shin (1987), and Samet (1987) have all drawn attention to this property. If the agent can recognize circumstances which confine the possible states of the world to R ∈ P ¯ , then whenever ω ∈ R , so that these circumstances do indeed obtain, the agent must realize that.

Definition The event E ⊂ Ω is self-evident to the agent who processes information according to P if P ( ω ) ⊂ E whenever ω ∈ E . A self-evident event can never occur without the agent knowing that it has occurred.

The axiom KTYK implies that every R ∈ P ¯ is self-evident to the agent.

Shin (1987) has suggested that KTYK and nondelusion are the only properties that need hold true for an agent whose knowledge was derived by logical deductions from a set of axioms.

Definition We say that P is nested if for all ω and ω ′ , either P ( ω ) ∩ P ( ω ′ ) = Ø , or else P ( ω ) ⊆ P ( ω ′ ) , or else P ( ω ′ ) ⊆ P ( ω ) .

An example might make the significance of nondelusion, KTYK, and nested clearer. Let there be just two propositions of interest in the universe, and let us suppose that whether each is true or false is regarded as good or bad, respectively. The state space is then Ω = { G G ,   G B ,   B G ,   B B } . One type of information processor P might always disregard anything that is bad, but remember anything that is good. Then P ( G G ) = { G G } , P ( G B ) = { G G ,   G B } , P ( B G ) = { G G ,   B G } , P ( B B ) = Ω . (See Figure 1a) It is clear that P satisfies nondelusion and KTYK, but does not satisfy nested. Moreover, when the reports are G B the agent chooses to remember only the first, while if they are B G he chooses to remember only the last. Alternatively, consider an agent with possibility correspondence Q who can remember G G and B B because the pattern is simple, and can also remember when he sees G B that the first report was good whereas with B G he remembers nothing. Then Q ( G G ) = { G G } , Q ( G B ) = { G B ,   G G } , Q ( B G ) = Ω , Q ( B B ) = { B B } . (See Figure 1b) This does satisfy nested, as well as the other two conditions. Nondelusion in these examples means that the agent never mistakes a good report for a bad report, or vice versa. KTYK means that if an agent recalls some collection of reports, then whenever all those reports turn out the same way he must also recall them (and possibly some others as well). Nested means that the reports are ordered in the agent's memory. If he remembers some report, then he must also remember every report that came earlier on the list.

We shall prove in Section 3 that nested can be interpreted as a property of memory in this way: Suppose that we think of a set S of fundamental propositions that can be either true or false. A state ω ∈ Ω specifies which of these propositions are true, and which are false. Suppose that knowledge at any ω can be described by a subset S ( ω ) ⊆ S . The agent knows at ω whether or not each proposition in S ( ω ) is true or false. In other words, P ω = ω ' ∈ Ω | s ∈ S ω ⇒ s  is true at  ω '   iff  s  is true at  ω . Finally, let us suppose that the propositions in S can be ordered (say chronologically) and that with respect to this ordering S ( ω ) is always an initial set, for any ω. Then P is nested (and nondeluded). Moreover, any nested and nondeluded P can be equivalently described this way. Nested corresponds to memory in the sense that the agent always remembers more or less far down the list of S, perhaps depending on how complicated the pattern of truth valuations is, but always in the same order.

Figure 1 shows that nondeluded and KTYK do not imply nested. Conversely, let Ω = { a ,   b ,   c } , and let P ( a ) = P ( c ) = { a ,   b ,   c } , while P ( b ) = { b ,   c } . Then P is nondeluded and nested, but P does not satisfy KTYK, since c ∈ P ( b ) but P ( c ) ⊆ P ( b ) .

Figure 1:

Non-deluded, KTYK, and nested illustrated.

2 Decision Theory Without Partitions

Our purpose in this paper is to analyze decision-making and game theory in environments where information processing is subject to error. Consider the following canonical decision problem:

Let A be a set of possible actions. Let u :   A × Ω → R . Let π be a measure^[1] on Ω . Let P :  Ω → 2 Ω be a possibility correspondence. We call a decision function f :  Ω → A optimal for the decision problem ( A ,  Ω ,   P ,   u ,   π ) iff

Condition (1) [ P ( ω ) = P ( ω ′ ) ] ⇒ [ f ( ω ) = f ( ω ′ ) ] .

Condition (2) For all ω ∈ Ω and a ∈ A ,

This definition applies for any possibility correspondence, whether or not it is a partition. Notice both conditions (1) and (2) serve to limit choices to reflect the level of information. Condition (1) requires that the agent's action is a function of what he perceives, and condition (2) requires that the agent optimizes, taking his information at face value.

In the above definition the agent is effectively unaware that he is erring in his information processing. Given the information that the state of nature ω ′ ∈ R ≡ P ( ω ) , the agent routinely uses Bayes Law to update his beliefs and to optimize.^[2] Were he aware of his errors, he would refine the possibility correspondence into a partition by letting P ˆ ( ω ) = { ω ′ ∈ Ω | P ( ω ′ ) = P ( ω ) } .

In the above decision framework the agent does not completely understand the model. He also does not necessarily “know what he is doing.” If he knew his optimal plan f :   Ω → A , and knew at each ω what his choice ought to be, then he would further refine his information according to the partition Q f ( ω ) = { ω ′ ∈ Ω | f ( ω ′ ) = f ( ω ) } . It might then turn out that his optimizing behavior would no longer correspond to f.

To illustrate the definition above we shall shortly present three examples which shall be of further use in later sections. At the same time we investigate the precise sense in which an agent who knows more but is boundedly rational may be worse off than if he knew less but was unboundedly rational.

A fundamental consequence of Bayesian decision making, and unbounded rationality, is that knowing more can never be disadvantageous. If P : Ω → 2 Ω and Q : Ω → 2 Ω then we say that Q is coarser than P if P ( ω ) ⊆ Q ( ω ) for all ω. If g is optimal for ( A ,   Ω ,   Q ,   u ,   π ) , and Q is coarser than P, and P and Q are partitions, then

Condition (3) ∑ ω ∈ Ω u ( g ( ω ) ,   ω ) π ( ω ) ≤ ∑ ω ∈ Ω u ( f ( ω ) ,   ω ) π ( ω ) .

It turns out that this property of Bayesian decisions is at the heart of the non-speculation literature. By allowing for less rational information processing it need no longer be the case that more knowledge is better.

In fact, one wonders if there are any general properties at all that can be proved outside the Bayesian framework. We shall show however that there are. Indeed the “more is better property” applies to a more general set of information correspondences than partitions.

Example 2.1 Let Ω = { a ,   b ,   c } , P ( a ) = { a ,   b } , P ( b ) = { b } , P ( c ) = { b ,   c } . (Note that ( Ω ,   P ) satisfies nondeluded and KTYK, but not nested.) Let π ( a ) = π ( c ) = 2 / 7 and π ( b ) = 3 / 7 . Let the action set be A = { B ,   N } , corresponding to bet or not bet. Let the payoffs from not betting be u ( N ,   a ) = u ( N ,   b ) = u ( N ,   c ) = 0 . Let the payoffs to betting be u ( B ,   a ) = u ( B ,   c ) = − 1 , while u ( B ,   b ) = 1 . It is easy to calculate that f ( ω ) = B for all ω ∈ Ω is optimal for ( A ,  Ω ,   P ,   u ,   π ) . Yet,

Of course g ( ω ) = N for all ω ∈ Ω is optimal for ( A ,  Ω ,   Q ,   u ,   π ) where Q ( ω ) = Ω for all ω, so for this example inequality (3) fails.

Example 2.2 Let Ω = { a ,   b ,   c } , and let P ( a ) = P ( c ) = { a ,   b ,   c } , while P ( b ) = { b ,   c } . Then ( Ω ,   P ) is nondeluded and nested, but does not satisfy KTYK. Let A = { B ,   N } , let π ( ω ) = 1 / 3 , for all ω ∈ Ω , and let u ( N ,   ω ) = 0 for all ω ∈ Ω , while u ( B ,   a ) = − 2 , u ( B ,   b ) = − 2 , u ( B ,   c ) = 3 . Then f = ( f ( a ) ,   f ( b ) ,   f ( c ) ) = ( N ,   B ,   N ) is optimal for ( A ,   Ω ,   P ,   u ,   π ) , but

Once again let the coarse partition be Q ( ω ) = Ω for all ω. Then g ( ω ) = N for all ω ∈ Ω is optimal for ( A ,   Ω ,   Q ,   u ,   π ) , and again (3) is violated.

Example 2.3 Let Ω = { a ,   b } , let P ( a ) = { a } , P ( b ) = { a ,   b } . Then ( Ω ,   P ) satisfies all these properties nondeluded, nested, and KTYK. Let A = { B ,   N } , let π ( a ) = π ( b ) = 1 / 2 . Let u ( N ,   a ) ≡ u ( N ,   b ) = 0 , let u ( B ,   a ) = 1 , and u ( B ,   b ) = − 2 . Then f ( a ) = B , f ( b ) = N is optimal for ( A ,  Ω ,   P ,   u ,   π ) , and

In this case condition (3) holds. Observe also that if we changed the payoff at ( B ,   b ) to u ( B ,   b ) = − 1 , then there would be a second optimal decision function f ˜ ( a ) = B = f ˜ ( b ) . In that case ∑ ω ∈ Ω u ( f ˜ ( ω ) ,   ω ) π ( ω ) = 0 is worse than the payoff arising from f, but still as good as the payoff arising from ( A ,   Ω ,   Q ,   u ,   π ) where Q ( ω ) = Ω for all ω ∈ Ω .

In Examples 2.1 and 2.2, the agent was (ex ante) worse off knowing more because he did not process information coherently. In Example 2.3 the agent was not worse off, although he also did not process information correctly. In general we have:

Conversely, suppose that ( Ω ,   P ) fails to satisfy one or more of the above hypotheses. Then there is a partition Q of Ω that is a coarsening of P, and an A, u, π such that f, g are optimal for ( A ,  Ω ,   P ,   u ,   π ) , ( A ,  Ω ,   Q ,   u ,   π ) , respectively, and yet the above inequality is strictly reversed.

Proof The proof of the first half of the theorem proceeds by induction on the cardinality of Ω . Suppose # Ω = 1 . Then P ( ω ) = Ω = Q ( ω ) for all ω ∈ Ω , and there is nothing to prove. Suppose the theorem is true for # Ω ≤ k . Consider the case where # Ω = k + 1 . Let S = { ω ∈ Ω :   P ( ω ) = Ω } . If S = Ω , then again there is nothing to prove. If S ≠ Ω , let Ω 1 = P ( ω ¯ ) be any possibility set with the greatest cardinality less than k + 1 . Then by nondelusion 0 < # Ω 1 ≤ k , and # Ω \ Ω 1 ≤ k . From knowing that you know, if ω ∈ Ω 1 , P ( ω ) ⊂ Ω 1 so S ∩ Ω 1 = Ø . Let Ω 2 = Ω \ Ω 1 ∪ S . From nondeluded and nested, if ω ∈ Ω 2 , P ( ω ) ∩ Ω 1 = ∅ . From KTYK, nondeluded, and the definition of S, if ω ∈ Ω 2 , P ( ω ) ∩ S = Ø . Hence, if ω ∈ Ω 2 , P ( ω ) ⊂ Ω 2 . Let I be the partition of Ω formed by the disjoint sets Ω 1 , Ω 2 , and S. Consider the partition Q ∗ = Q ∨ I , defined by Q ∗ ( ω ) = Q ( ω ) ∩ I ( ω ) for all ω ∈ Ω . Let g * be optimal for ( A ,   Ω ,   Q ∗ ,   u ,   π ) . Then because Q and Q * are partitions, and Q is coarser than Q * on Ω 1 ∪ Ω 2 ,

But now we can apply the induction hypothesis to ( A ,   Ω 1 ,   P ,   u ,   π ) and ( A ,   Ω 2 ,   P ,   u ,   π ) , obtaining

Finally, let us observe that if S = Ø , we are finished. If there is some ω ˆ ∈ S , then Q is the trivial partition and we can assume WLOG that g ( ω ) = f ( ω ˆ ) for all ω ∈ Ω , and in particular g ( ω ) = f ( ω ) for all ω ∈ S . This concludes the proof of the first half of Theorem 1. The converse follows from Examples 2.1 and 2.2 by taking π (ω) = 0 for all but 3 elements of Ω.

We conclude this section by noting one important extension to decision theory that fits naturally into our framework. Let A ¯ be a correspondence specifying for each ω ∈ Ω the set of possible actions in some ambient space A, A ¯ :  Ω → 2 A . Then we would regard u as a function on A × Ω , and an optimal decision plan for ( A ¯ ,   A ,   Ω ,   P ,   u ,   π ) would be a function f :  Ω → A satisfying

Condition (1′) [ P ( ω ) = P ( ω ′ )  and  A ¯ ( ω ) = A ¯ ( ω ′ ) ] ⇒ [ f ( ω ) = f ( ω ′ ) ]

Condition (2′a) f ( ω ) ∈ A ¯ ( ω ) for all ω ∈ Ω and

Condition (2′b) For all ω ∈ Ω and a ∈ A ¯ ( ω )

This new formulation allows us to model the idea that agents take the face value of a message, and restrict their choices accordingly, without using the subtle content of the message (i. e., without using knowledge of the function A ¯ to invert the signal and so to deduce more about the state). We shall return to this theme in Section 10.

Corollary 1.1 Let ( A ¯ ,   A ,   Ω ,   P ,   u ,   π ) be a decision problem with variable cons-traints. Let ( Ω ,   P ) satisfy nondeluded, nested, and KTYK. Let [ P ( ω ) = P ( ω ′ ) ] ⇒ [ A ¯ ( ω ) = A ¯ ( ω ′ ) ] . Let Q be a partition of Ω that is a coarsening of P, and let A ˆ ( ω ) ⊂ A ¯ ( ω ) for all ω satisfy [ Q ( ω ) = Q ( ω ′ ) ] ⇒ [ A ˆ ( ω ) = A ˆ ( ω ′ ) ] . If f, g are optimal for ( A ¯ ,   A ,   Ω ,   P ,   u ,   π ) and ( A ˆ ,   A ,   Ω ,   Q ,   u ,   π ) , respectively, then ∑ ω ∈ Ω u ( g ( ω ) ,   ω ) π ( ω ) ≤ ∑ ω ∈ Ω u ( f ( ω ) ,   ω ) π ( ω ) .

Proof The proof is exactly as for Theorem 1.

3 Equivalent Decision Problems

Evidently the naming of states is somewhat arbitrary. For example, splitting a state into two indistinguishable states, which are physically identical but each with half the probability, should not change the decision problem. By formulating several definitions of equivalent decision problems we can clarify the framework of Section 2.

Definition We say that the decision problem ( A ,   Ω ′ ,   P ′ ,   u ′ ,   π ′ ) is a renaming of the decision problem ( A ,   Ω ,   P ,   u ,   π ) in the following senses:

Decision-theoretic, if there is a 1–1 and onto map δ :   P ′ ¯ → P ¯ such that for all R ′ ∈ P ′ ¯ , if R = δ ( R ′ ) then π ( R ) > 0 if and only if π ′ ( R ′ ) > 0 and if both are positive, then for all a ∈ A ,

Physical, if there is a function φ :   Ω ′ → Ω that is onto and satisfies

We say that two decision problems are equivalent (in either of the two senses) if they have a common renaming. An easy argument shows that these are indeed equivalence relations, and that a physical renaming is also a decision-theoretic renaming.

The decision theoretic equivalence notion was formulated by Brandenburger, Dekel, and Geanakoplos 1992. If it holds and if agents always optimize according to (1) and (2), then behaviorally the decision problems are equivalent. The following lemma also appears in (Brandenburger, Dekel, and Geanakoplos 1992):

Lemma 1 For any decision problem ( A ,  Ω ,   P ,   u ,   π ) there is a decision-theoretic renaming ( A ,   Ω ′ ,   P ′ ,   u ′ ,   π ′ ) in which P ′ is a partition of Ω ′ .

Sketch of Proof Let Ω ′ = P ¯ × Ω . Define u ′ :   A × Ω ′ → R by u ′ ( a ,   ( R ,   ω ) ) = u ( a ,   ω ) , P ′ ( R ,   ω ) = { R } × Ω , and δ ( { R } × Ω ) = R , and let

The upshot of Lemma 1 is that we can understand the information processing and decision problem of Section 2 as if the agent is a conventional maximizer, but has got the prior (on Ω ' ) wrong. The “correct” priors would be those that would obtain if the agent knew the function P, namely

The lemma thus provides us with another interpretation of decision theory with generalized partitions. If the reader wished, he could rewrite all of our results in terms of the consequences of using the wrong priors (or in later sections, of players using different priors). The advantage of the generalized possibility approach is that it explains how the mistaken priors might have arisen. Theorem 1, for example, gives conditions under which the agent does at least as well as he would with the right priors but less information. One perhaps could reformulate the result directly in terms of priors, but nondelusion, KTYK, and nested make clear just what information processing errors are tolerable.

We illustrate Lemma 1 with the ozone example in which Ω = { a ,   b } , π ( a ) = π ( b ) = 1 / 2 , P ( a ) = { a } , P ( b ) = { a ,   b } . Let γ = { a } correspond to “gamma rays” and n = { a ,   b } correspond to no gamma rays. Then Ω ′ = P ¯ × Ω = { γ a ,   γ b ,   n a ,   n b } , P ′ ( γ a ) = P ′ ( γ b ) = { γ a ,   γ b } , P ′ ( n a ) = P ′ ( n b ) = { n a ,   n b } , and π ′ ( γ a ) = 1 / 2 , π ′ ( γ b ) = 0 , π ′ ( n a ) = π ′ ( n b ) = 1 / 2 . The correct priors are π * ( γ a ) = 1 / 2 , π * ( γ b ) = π * ( n a ) = 0 , and π * ( n b ) = 1 / 2 . Note that the only partition of Ω weaker than P is the trivial partition, which under the transformation of Lemma 1 becomes the trivial partition Q ′ of Ω ′ . Theorem 1 can be interpreted to mean that any optimal plan f :   Ω ′ → A with respect to the prior π ′ and the partition P ′ does at least as well, evaluated according to the correct priors π * , as any plan feasible (i. e., measurable) with respect to the partition Q ′ .

Let us now introduce the idea of a particularly simple, concrete description of the set of states of nature. Let us call Ω a propositional state space if Ω = 2 n . Each  ω ∈ Ω can be interpreted as a truth assignment to each of n ordered propositions, and we can represent ω as an n-tuple of binary digits: ω = ( ω ( 1 ) ,   … ,   ω ( q ) ,   … ,   ω ( n ) ) . We call A ⊂ 2 n a basic propositional event if there is some proposition q such that A corresponds to all states assigning the same truth valuation to q. More precisely, there is ω ¯ ∈ Ω such that A = { ω ∈ Ω | ω ( q ) = ω ¯ ( q ) } . A propositional event A ⊂ Ω is a nonempty intersection of basic propositional events: there exists ω ¯ ∈ Ω and 1 ≤ q 1 ≤ ⋯ ≤ q m ≤ n such that A = { ω ∈ Ω | ω ( q i ) = ω ¯ ( q i ) ,   i = 1 ,   … ,   m } .

Notice that so far the ordering of the propositions has not played an essential role in our definitions. We suggest that a property of memory is that the propositions (or basic propositional events) are arranged in some definite order in the mind, perhaps from most important to least important, or reverse chronologically from most recent to most distant, so that one remembers the outcomes in order. Sometimes one might remember more or less (perhaps depending on the complexity of the outcomes), but always in the same order. More precisely:

Definition Let P ¯ be a collection of subsets of a propositional state space Ω = 2 n . We say that P ¯ has the memory property if for any R ∈ P ¯ , there is ω ¯ ∈ Ω and 0 ≤ k ≤ n such that R = { ω ∈ Ω | ω ( q ) = ω ¯ ( q ) ,   q = 1 ,   … ,   k } .

We now show that nested can always be interpreted in terms of memory.

Lemma 2 Let ( A ,  Ω ,   P ,   u ,   π ) be given and let ( Ω ,   P ) be nondeluded and nested. Then there exists a physical renaming ( A ,   Ω ′ ,   P ′ ,   u ′ ,   π ′ ) which is propositional and has the memory property.

Proof The proof proceeds by induction on # Ω . For Ω = { ω } , let there be one proposition n = 1 and let Ω ′ = { T ,   F } , let P ′ ( T ) = P ′ ( F ) = Ω ′ , and let u ′ ( a ,   T ) = u ′ ( a ,   F ) = u ( a ,   ω ) for all a ∈ A . Finally, let π ′ ( T ) = π ′ ( F ) = 1 2 π ( ω ) .

Suppose now that the lemma is true for # Ω ≤ k , and let # Ω = k + 1 . Let T 0 = { ω ∈ Ω | ω ∈ R ∈ P ¯ only if R = Ω } . If T 0 = Ω , there is nothing to prove. So suppose # T 0 ≤ k . For each ω ∈ Ω\ T 0 , let T ( ω ) be the set in P ¯ with largest cardinality ≤ k containing ω. Then by nested, ( T 0 ,   { T ( ω ) | ω ∈ Ω \ T 0 } ) is a non-trivial partition of Ω . By combining sets of the form T(ω), we may suppose that the partition T consists of two sets, each containing at most k states. For any T i , and ω ∈ T i , by nondeluded, either P ( ω ) = Ω or P ( ω ) ⊂ T i . Let P i :   T i → 2 T i \ { Ø } be defined by P i ( ω ) = P ( ω ) ∩ T i . Then each of the decision problems ( A ,   T i ,   P i ,   u ,   π ) satisfies the induction hypothesis. Hence there are physical renamings A ,  Ω i ′ ,   P i ′ ,   u i ′ ,   π i ′ where each Ω i ′ = 2 n i . To every ω i ∈ Ω i ′ there is an integer 0 ≤ k i ( ω i ) ≤ n i such that P i ′ ω i = ω ∈ Ω i ′ | ω q = ω i q ,   q = 1 ,   … ,   k i ω i . By adding irrelevant propositions (whose truth is never distinguished by the P i ′ ) WLOG  n 2 = n 1 and Ω 2 ′ ≈ Ω 1 ′ = 2 n 1 . Let ( A ,   Ω ′ ,   P ˜ ′ ,   π ′ ,   u ′ ) be defined by Ω ′ = Ω 1 ′ ∪ Ω 2 ′ = 2 × 2 n 1 . P ′ ˜ is defined by P ′ ˜ ( ω ′ ) = P i ( ω ′ ) if ω ′ ∈ Ω i ′ ; π ′ is defined by π ′ ( ω ′ ) = π i ( ω ′ ) if ω ′ ∈ Ω i ′ ; u ′ is defined by u ′ ( ω ′ ) = u i ( ω ′ ) if ω ′ ∈ Ω i ′ . It is clear that ( A ,   Ω ′ ,   P ˜ ′ ,   π ′ ,   u ′ ) is a physical renaming with the memory property for ( A ,  Ω ,   P ˜ ,   π ,   u ) , where P ˜ ( ω ) = P i ( ω ) if ω ∈ T i . (Each P ˜ ′ ( ω ′ ) is characterized by specifying a truth valuation to one proposition distinguishing whether ω ′ ∈ Ω 1 ′ or ω ∈ Ω 2 ′ , to k i ( ω ′ ) of the next n 1 propositions if ω ′ ∈ Ω i ′ ). Finally, recall that P ˜ ( ω ) = P i ( ω ) differs from P ( ω ) only if P ( ω ) = Ω . Define P ′ on Ω ′ = Ω 1 ′ ∪ Ω 2 ′ , as follows:

Clearly, if P ′ ( ω ′ ) = Ω ′ , then this set can be characterized by specifying the truth valuation of none of the 1 + n 1 propositions. Hence ( A ,   Ω ′ ,   P ′ ,   u ′ ,   π ′ ) is a physical renaming of ( A ,   Ω ,   P ,   u ,   π ) that has the memory property.

4 Game Theory with Generalized Partitions

We can extend game theory, as well as decision theory, to allow for information processing errors. The fundamental notion in game theory is the equilibrium concept provided by John Nash in 1951. Let G = ( I ,   A i ,   Ω ,   P i ,   u i ,   π i ) , i = 1 , … , I , be a collection of decision makers. Let u i have domain X i = 1 I A i × Ω . Then we say that the functions ( f i ,   i = 1 ,   … ,   I ) are a Nash equilibrium for the Game G iff

1. [ P i ( ω ) = P i ( ω ′ ) ] ⇒ [ f i ( ω ) = f i ( ω ′ ) ] , ω, ω ′ ∈ Ω , i ∈ I .

2. For all ω ∈ Ω , i ∈ I , and a i ∈ A i ,

We can interpret our definition of Nash equilibrium in much the same way as we did the single agent decision-maker. The players with non-partional P_i do not completely understand the model, and so they are led to make information processing blunders concerning the signals they receive.

One consequence of this point of view is that one of the rationalizations for Nash equilibrium, that each player deduces what he should do as a matter of logical introspection, is no longer tenable. However, many of the other interpretations of equilibrium are still viable. For example, an equilibrium can still be characterized as an agreement from which no agent has an incentive to deviate.

Let us emphasize one limitation of the current model. Agents are permitted to make errors about the significance of their signals. But these errors do not depend on the moves of other agents, even though, for example, the news that some state has occurred might be much more unpleasant depending on what the other players plan to do in that state. If we extended our equilibrium notion to allow for correlated equilibria, as in (Brandenburger, Dekel, and Geanakoplos 1992), then it would be natural to allow the errors to depend on the moves of other agents. (e. g., there might be some things that you simply refuse to believe somebody else would ever do.)

We now give three examples illustrating the definition of Nash equilibrium with generalized partitions, which will also serve to introduce the idea of speculation.

Example 4.1 Let I = { 1 ,   2 } . Let Ω = { a ,   b ,   c } , P 1 ( a ) = { a ,   b } , P 1 ( b ) = { b } , P 1 ( c ) = { b ,   c } . Let P 2 ( ω ) = Ω for all ω ∈ Ω . Let π 1 = π 2 = π , with π ( a ) = π ( c ) = 2 / 7 , π ( b ) = 3 / 7 . Let the action spaces be A 1 = A 2 = { B ,   N } . Finally, let the payoffs in the three states be:

where ϵ > 0 is small. It is clear that there are two Nash equilibria. In the first, f i ( ω ) = N , ∀ ω ∈ Ω , i = 1 ,   2 . In the second, f i ( ω ) = B , ∀ ω ∈ Ω , i = 1 ,   2 . We are most interested in the possibility of the second equilibrium. Here the agents always bet against each other, simply on account of different information. Indeed, although agent 1 always knows strictly more than agent 2, on account of his “irrational” (generalized partition) information processsing, on average he is losing money.

Example 4.2 Let I = { 1 ,   2 } . Let Ω = { a ,   b ,   c } , P 1 ( a ) = P 1 ( c ) = Ω , P 1 ( b ) = { b ,   c } . Let P 2 ( ω ) = Ω for all ω ∈ Ω . Let π 1 ω = π 2 ω = π ω = 1 3 for all ω ∈ Ω . Let the action spaces be A 1 = A 2 = { B ,   N } . Finally, let the payoffs in the three states be:

where ϵ > 0 is small. Again there are two Nash equilibria. In the trivial one, f i ( ω ) = N , ∀ ω ∈ Ω , i = 1 ,   2 . In the second, f 1 a = f 1 c = N , but f 1 ( b ) = B , f 2 ( ω ) = B , ∀ ω ∈ Ω . Again, in the interesting equilibrium a bet does take place, though not always. Note also that agent 2 is willing to bet (always) because he knows that the only time his bet will be taken up is in state b, where he wins. Agent 1 knows that 2 is always willing to bet, but does not realize that he himself only bets when he is sure to lose. Once again agent 1 loses out to agent 2 despite his superior knowledge at each ω, because he is not perfectly rational.

Example 4.3 Let I = { 1 ,   2 } . Let Ω = { a ,   b } , P 1 ( a ) = { a } , P 1 ( b ) = { a ,   b } , P 2 ( a ) = P 2 ( b ) = Ω . Let π ( ω ) = π i ( ω ) = 1 / 2 , i = 1 ,   2 , ω ∈ Ω . Let the action spaces be A 1 = A 2 = { B ,   N } , and the payoffs in the two states be: