Dynamic Variations: Update and Revision for Diverse Agents

characterization of product update Consider a tree-like structure E with possible events (or actions) and uncertainty relations among its nodes, which can also verify atomic propositions p, q, ... The only contrast with a real tree is that we allow a bottom level with multiple roots. Nodes X, Y, ... are at the same time finite sequences of events, and the symbol ∩ expresses concatenation of events. Intuitively, we think of such a tree structure E as the possible evolutions of some process – for instance, a game. A particular case is the above model Tree(M , A) starting from an initial epistemic model M and an action model A, and repeating product updates forever. Now, the preceding discussion shows that the following two principles are valid in Tree(M, A), which can be stated as general properties of a tree E . They represent Perfect Recall and ‘Uniform No Learning’, respectively: 54 CHAPTER 3. DIVERSITY OF LOGICAL AGENTS IN GAMES PR If X∩(a) ∼iY, then ∃b ∃Z: Y = Z ∩(b) & X ∼i Z. UNL If X∩(a) ∼i Y ∩(b), then ∀U, V: if U ∼iV, then U ∩(a) ∼i V ∩(b), provided that U∩(a), V ∩(b) both occur in the tree E . Moreover, the special nature of the preconditions in product update, as definable conditions inside the current epistemic model, validates one more abstract constraint on the tree E : BIS-INV The set {X | X∩(a) ∈ E} of nodes where action a can be performed is closed under purely epistemic bisimulations of nodes. Now we have all we need to prove a converse representation result. Theorem For any tree E , the following are equivalent: (a) E ∼= Tree(M , A) for some M, A (b) E satisfies PR, UNL, BIS-INV Proof From (a) to (b) is the above observation. Now, from (b) to (a). Define an epistemic model M as the set of initial points in E and copy the relations ∼i from E . The action model A contains all possible actions occurring in the tree, where we set a ∼i b iff ∃X ∃Y: X ∩(a) ∼i Y ∩(b) We also need to know that the preconditions PREa for actions a are as required. For this, we use the well-known fact that in any epistemic model, any set of worlds that is closed under epistemic bisimulations must have a definition in the epistemic language – though admittedly, one allowing infinite conjunctions and disjunctions. The abstract setting of our result allows no further finitization of this definability. Now, the obvious identity map F sends nodes X of E to corresponding states in the model Tree (M ,A). First, we observe the following fact about E itself: Lemma If X∼iY, then length(X) = length(Y). 3.3. UPDATE FOR PERFECT AGENTS 55 Proof If X, Y are initial points in E , both their lengths are 0. Otherwise, let X have length n+1. By PR, X ’s initial segment of length n stands in the relation ∼i to a proper initial segment of Y whose length is that of Y minus 1. Repeating this observation peels off both sequences to initial points after the same number of steps. Claim X ∼i Y holds in E iff F(X) ∼i F(Y) holds in Tree(M , A). The proof is by induction on the common length of the two sequences X, Y. The case of initial points is clear by the definition of M. As for the inductive steps, consider first the direction ⇒. If U∩(a) ∼iV, then by PR, ∃b ∃Z: V = Z∩(b) & U ∼i Z. By the inductive hypothesis, we have F(U) ∼i F(Z). We also have a ∼i b by the definition of A. Moreover, given that the sequences U∩(a), Z∩(b) both belong to E , their preconditions as listed in A are satisfied. Therefore, in Tree(M , A), by the definition of product update, (F(U), a) ∼i (F(Z), b), i.e. F(U ∩(a)) ∼i F(Z ∩(b)). As for the direction ⇐, suppose that in Tree(M , A) we have (F(U), a) ∼i (F(Z), b). Then by the definition of product update, F(U) ∼i F(Z) and a ∼i b. By the inductive hypothesis, from F(U) ∼i F(Z) we get U ∼i Z in E(*). Also, by the given definition of a ∼i b in the action model A, we have ∃X ∃Y: X∩(a) ∼i Y ∩(b)(**). Taking (*) and (**) together, by UNL we get U∩(a) ∼i Z ∩(b), provided that U∩(a), V ∩(b) ∈ E . But this is so since the preconditions PREa, PREb of the actions a, b were satisfied at F(U), F(Z). This means these epistemic formulas must also have been true at U, V – so, given what PREa, PREb defined, U ∩(a), V ∩(b) exist in the tree E .¥ This result is only one of a kind, and its assumptions may be overly restrictive. In many game scenarios, preconditions for actions are not purely epistemic, but rather depend on what happens over time. E.g., a game may have initial factual announcements – like the Father’s saying that at least one child is dirty in the puzzle of the Muddy Children. These are not repeated, even though their preconditions still hold at later stages. Describing this requires preconditions PREa for actions a that refer to the temporal structure of the tree E , and then the above invariance for purely epistemic bisimulations would fail. Another strong assumption is our use of a single action model A that gets repeated all the time in levels M, (M×A), (M×A)×A, ... to produce the structure Tree(M, A). A more local perspective would allow different action models A1,A2, ... in stepping from 56 CHAPTER 3. DIVERSITY OF LOGICAL AGENTS IN GAMES one tree level to another. And an even more finely-grained view arises if single moves in a game themselves can be complex action models. In the rest of this paper, for convenience, we stick to the single-model view. 3.4 Update Logic for Bounded Agents Limitations on information processing The information-processing capacity of agents may be bounded in various ways. One of these is ‘external’: Agents may have restricted powers of observation. This kind of restriction is built into the definition of action models, with uncertainties for agents – and the product update mechanism of Section 3.2 reflects this. Another type of restriction is ‘internal’: Agents may have bounded memory. Agents with Perfect Recall had limited powers of observation but perfect memory. At the opposite extreme we find Memory-free agents which can only observe the last event, without maintaining any record of what went on before. In this section, we explore this extreme case. Characterizing types of agent In the preceding, agents with Perfect Recall have been described in various ways. Our general setting was the tree E of event sequences, where different types of agents i correspond to different types of uncertainty relation ∼i. One approach was via structural conditions on such relations, such as PR, UNL, and BIS-INV in the above characterization theorem. Essentially, these three constraints say that X ∼i Y iff length(X) = length(Y) and X(s) ∼i Y(s) for all positions s Next, these conditions also validated corresponding axioms in the dynamicepistemic language that govern typical reasoning about the relevant type of agent. But thirdly, we also think of agents as a sort of processing mechanism. Intuitively, an agent with Perfect Recall is a push-down store automaton maintaining a stack of all past events and continually adding new observations. Such a processing mechanism was provided by our representation theorem, viz. epistemic product update. Bounded memory Another broad class of agents arises by assuming bounded memory up to some fixed finite number k of positions. In general trees E , this makes two event sequences X, Y ∼i-equivalent for such agents 3.4. UPDATE LOGIC FOR BOUNDED AGENTS 57 i iff their last k positions are ∼i-equivalent. In this section we only consider the most extreme case of this, viz. Memory-free agents i : X ∼i Y iff last(X) ∼i last(Y) or X = Y = the empty sequence $ Agents of this sort only respond to the last-observed event. In particular, their uncertainty relations can now cross between different levels of a game tree: They need not know how many moves have been played. Perhaps contrary to appearances, such limited agents can be quite useful. Examples are Tit-for-Tat players in the iterated Prisoner’s Dilemma which merely repeat their opponents’ last move (Axelrod 1984), or Copy-Cat players in game semantics for linear logic which can win ‘parallel disjunctions’ of games G ∨ G (Abramsky 1996). Incidentally, these are players with a hard-wired strategy : a point that we will discuss below. It is easy to characterize such agents in terms similar to what we did with Perfect Recall. Fact An equivalence relation ∼i on E is Memory-free in the sense of $ if and only if the following two conditions are satisfied: PR− If X∩(a) ∼i Y, then ∃b ∼i a ∃Z: Y = Z ∩(b). UNL+ If X∩(a) ∼i Y ∩(b), then ∀U, V: U∩(a) ∼i V ∩(b), provided that U∩(a), V ∩(b) both occur in the tree E . Proof If an agent i is Memory-free, its relation ∼i evidently satisfies PR − and UNL+. Conversely, suppose that these conditions hold. If X ∼i Y, then either X, Y are both the empty sequence, and we are done, or, say, X = Z(a). Then by PR−, Y = U(b) for some b ∼i a, and so last(X) ∼i last(Y). Conversely, the reflexivity of ∼i plus UNL + imply that, if the right-hand side of the equivalence $ holds, then X ∼i Y. ¥ It is also easy to give a characteristic modal-epistemic axiom for this case. First, set a ∼i b iff ∃X ∃Y : X ∩(a) ∼i Y ∩(b) Fact The following equivalence is valid for Memory-free agents: 〈a〉〈i〉φ ↔ (PREa & E ∨