Information, Beliefs, Knowledge and Rationality in Games

The Standard State Space Models The Common Prior Assumption Approximate Common Knowledge Epistemic Foundation for Normal-Form Games Epistemic Foundation for Extensive-Form Games The State Space Model Revisited

Topics in Foundations of Game Theory Information, Beliefs, Knowledge and Rationality in Games

Jing Li

Department of Economics Duke University

February 21, 2007

Jing Li Topics in Foundations of Game Theory The Standard State Space Models The Common Prior Assumption The Basic Structure Approximate Common Knowledge Representing Knowledge Epistemic Foundation for Normal-Form Games Common Knowledge Epistemic Foundation for Extensive-Form Games Agreeing to Disagree The State Space Model Revisited

The Primitives: (Ω, {Pi}i∈I)

The set representation of uncertainty. Ω: the state space, an arbitrary set E = {E : E ⊆ Ω}: set of events; e.g. Ω, ∅; representing logical relations using set operations: E ⊆ F (E logically implies F); ∩ (and); ∪ (or); Ω \ E (not E). The information structure. Ω Pi :Ω → 2 \ {∅} , i ∈ I: the possibility correspondence Pi(ω): the set of states i considers possible at ω; interpreted as the information i receives at s;

The model (Ω, {Pi}i∈I) itself is “common knowledge.” (Aumann, Harsanyi 1967/68, Mertens and Zamir 1985, Brandenburger and Dekel 1993, Bacharach 1985)

Pi induces an information partition if: 1 P1: ω ∈ Pi(ω) for all ω ∈ Ω; 0 0 2 P2: ω ∈ Pi(ω) ⇒ Pi(ω ) ⊆ Pi(ω); 0 0 3 P3: ω ∈ Pi(ω) ⇒ Pi(s ) ⊇ Pi(ω).

The model is partitional if Pi induces an information partition for all i; otherwise it is non-partitional. Justiﬁcation for information partition. P0(ω) = {ω0 : P(ω0) = P(ω)}; Information is part of the speciﬁcation of the state, and P is simply a coding system.

Deﬁne the knowledge operator Ki : E → E

Ki(E) = {ω ∈ Ω: Pi(ω) ⊆ E} . (1.1)

Truth in all possible states; Use iterations to capture higher-order interactive knowledge; What’s in a state: ··· + everybody’s knowledge hierarchy; Higher-order interactive knowledge: iterating the knowledge operators.

Basic properties of knowledge: hold for arbitrary P; K0a: K(Ω) = Ω; K0b: E ⊆ F ⇒ K(E) ⊆ K(F); K0c: K(E) ∩ K(F) = K(E ∩ F); Rational properties of knowledge: corresponding to P1 − 3; K1: K(E) ⊆ E; K2: K(E) ⊆ KK(E); K3: ¬K(E) ⊆ K¬K(E).

The knowledge hierarchies are pinned down at the ﬁrst level. For any event E ⊆ Ω,

Ki(E) ∪ ¬Ki(E) = Ω for all i;

Ki(E) = KiKi(E) for all i;

¬Ki(E) = Ki¬Ki(E) for all i;

KiKj(E) ⊆ Kj(E) for all i, j;

An event E is common knowledge if everyone knows E, and everyone knows that everyone knows E, and everyone knows that everyone knows everyone knows E, and so on. The natural deﬁnition:

“Everybody knows” operator: K∗(E) = ∩i∈IKi(E); ∞ m CK(E) = ∩m=1K∗ (E).

The ﬁxed point characterization:

An event G is self-evident if G ⊆ K∗(G); ω ∈ CK(E) if and only if there exists a self-evident event G such that G ⊆ E. Characterization in partitional models:

n CK(E) = {ω ∈ Ω: ∧i=1Pi(ω) ⊆ E} (1.2)

Note: CK(E) = CK(CK(E)).

Primitives: a probability space (Ω, E, π1, π2), information partitions P1, P2;

Knowledge K1, K2 as deﬁned in (1.1);

Result: Suppose π1 = π2 and π(ω) > 0 for all ω ∈ Ω. Then for any E ⊆ Ω, if πi(E|Pi(ω)), i = 1, 2 are common knowledge at ω, then π1(E|P1(ω)) = π2(E|P2(ω)).

Example 1: The muddy face puzzle; Example 2: The detective story; Example 3: reaching agreement on posterior beliefs; We can’t disagree for ever: under common prior and ﬁnite information partitions, players reach agreement in ﬁnite steps through direct communication. (Geanakoplos and Polemarchakis 1982)

Primitives: a probability space (S = Θ × X, E, {πi}i∈I), and for each i ∈ I, an information partition Pi (on the signal space X), an initial endowment ei :Θ → R, a utility function ui :Θ × R → R; Knowledge Ki as deﬁned in (1.1); n A feasible trade is a function t :Θ → R that satisﬁes: P Budget balance: i∈I ti ≤ 0; No starvation: ei(θ) + ti(θ) ≥ 0.

Result: Suppose players are risk-averse. If πi(·|Y) = πj(·|Y) for all Y ⊆ X and i, j ∈ I, and the initial endowment e = (e1, ··· , en) is Pareto optimal, then if it is common knowledge that t∗ is a feasible trade, then t∗ is essentially the null trade.

Main message: private information alone cannot induce trade. Key components: Ex-post common knowledge forces learning, which helps to symmetrize private information; The learning depends on the content of common knowledge, and the assumptions linking such contents with information, for example, rationality assumption; Non-partitional information structures would put some restrictions on learning (but not much); The common prior assumption. The logic of no-trade type results: Rubinstein and Wolinsky 1990

Jing Li Topics in Foundations of Game Theory The Standard State Space Models The Common Prior Assumption Key Issues Approximate Common Knowledge The Universal Type Space Epistemic Foundation for Normal-Form Games The CPA in Games Epistemic Foundation for Extensive-Form Games Characterizing the CPA The State Space Model Revisited The Concept of Probability

Objective view: intrinsic physical property of the random phenomenon; unknown, ﬁxed truth. Subjective view: personal beliefs satisfying certain mathematical properties; “consistent” choice under uncertainty (Savage). Bayesian convergence theorem: after observing sufﬁcient data, all prior probability distributions converge to the same limit, and the limit is a dirac measure.

Genuine prior stage: prior over the set of states of nature; The type space in games of incomplete information. The type space is merely a representation of the inﬁnite posterior belief hierarchies; Is there a meaningful prior stage? The only way to understand the CPA in this environment is to understand it through posterior belief hierarchies.

Approach: Given a set of states of nature S, construct belief hierarchies explicitly. ∞ The space of all types: T0 = ×n=0∆(Xn) X0 = S; X1 = X0 × ∆(X0) ··· Xn = Xn−1 × ∆(Xn−1) ··· i i i ∞ t = (δ1, δ2, ··· ) ∈ ×n=0∆(Xn), Coherency: The different levels of beliefs of an individual do not

contradict one another, i.e. margXn−2 δn = δn−1. The space of coherent types: T1 = t ∈ T0 : margXn−2 δn = δn−1

Coherency: The different levels of beliefs of an individual do not

contradict one another, i.e. margXn−2 δn = δn−1. The space of coherent types: T1 = t ∈ T0 : margXn−2 δn = δn−1 Lemma: There exists a homeomorphism f : T1 → ∆(S × T0). Interpretation: when restricting to coherent types, there is no need to go to second-level beliefs over the opponents’ type spaces. If a type is coherent, it assigns unit mass to a probability measure over others’ type spaces.

Deﬁne knowledge as belief with probability 1: type t knows a measurable event F in the space S × T0 if f (t)(F) = 1;

Type t knows the opponent is a coherent type if f (t)(S × T1) = 1; Common knowledge of coherency:

Tk = {t ∈ T1 : f (t)(S × Tk−1) = 1}; t ∈ Tk: type t knows the opponent knows ··· knows the opponent is a coherent type (k − 1 times); ∞ T = ∩k=1Tk; The set of types satisfying common knowledge of coherency: T × T. Result: There is a homeomorphism g : T → ∆(S × T).

The self-referential problem of the knowledge structure revisited: in what sense is the model (Ω, P) common knowledge? Ω = S × T × T; (the size issue) Type t knows precisely his own type and hence has conditional probability: for a measurable event F in S × T × T, his belief is g(t)({(s, t0):(s, t, t0) ∈ F}); Given event E ⊆ S, the event “i knows E is deﬁned by: 0 Ki(E) = {(s, t, t ): g(t)(E × T) = 1}; Result: Given any event E ⊆ S, the two deﬁnitions are equivalent. Common knowledge of (Ω, P) is tantamount to common knowledge of coherency.

Learning leads to identical beliefs; Without the CPA, there is too much ﬂexibility and hence weak predictability; The CPA allows one to separate informational issues: Difference in beliefs should reﬂect only difference in private information. Pragmatic value: tractability;

A strong assumption if imposed on truly “prior” stage: No higher-order uncertainties; Identical personal beliefs regarding fundamentals. Problematic if imposed in games of incomplete information, or in epistemic analysis of games: There is typically no meaningful prior stage; The CPA can only be understood through its restrictions on posterior belief hierarchies; The CPA imposes restrictions on posterior belief hierarchies, and not necessarily jointly; Gul’s example.

Learning opportunities are typically rare; “Without the CPA, everything can happen” may not be true; (Morris 1994) Bottom line: The CPA assumes (much) more than just “difference in probabilities should reﬂect difference in private information.”

The type space: Ω

The type function ti :Ω → ∆(Ω); The (non)-existence of CP is equivalent to separation of two convex sets; (Samet 1998b)

The set of “priors” for player i: Pi = co {ti(ω): ω ∈ Ω} The existence of common prior is equivalent to ∩i∈IPi 6= ∅. The existence of CP is equivalent to the existence of a common invariant probability measure for Markov chains; (Samet 1998a) 0 0 The Markov transition matrix for player i: Mi(ω, ω ) = ti(ω)(ω ); For any random variable f on Ω, Mif =< Ei(f |Pi(ω)) >ω∈Ω; A probability measure p over Ω is CP iff pMi = Mi for all i.

In finite models, the converse of Aumann’s result holds: the CPA is equivalent to no common knowledge of disagreement. Various expressions of the posterior characterization: The iterated expectations of any random variable converge to the same limit, which is equal to the expectation under the common prior; (Samet 1989b) No CPA iff players disagree on the odds of some natural event combined with the outcome of finite tosses of a fair coin; (Feinberg 2000) No CPA iff Players “disagree” on the odds of a finite collection of natural events; (Heifetz 2003); The implications of the CPA in infinite models are not well understood. Exceptions: Halpern 2002, Lipman 2005.

The posterior characterization of the CPA in finite models always involves common knowledge, i.e. a statement about infinite orders of beliefs; In finite models, up to any finite order beliefs, the CPA is equivalent to the assumption of common support; (Lipman 2003) Implications: Whether the CPA holds can only be checked via infinite orders of beliefs; The main force of the CPA is the requirement the prior be common knowledge, i.e. infinite order of mutual knowledge; Any result crucially dependent on the CPA must also crucially depend on common knowledge.

Jing Li Topics in Foundations of Game Theory The Standard State Space Models Approximate the Order The Common Prior Assumption Approximate Knowledge Approximate Common Knowledge Common p-Beliefs Epistemic Foundation for Normal-Form Games Robust Equilibrium Epistemic Foundation for Extensive-Form Games Global Games The State Space Model Revisited Rubinstein’s E-mail Game

Rubinstein 1989 Set-up: simple coordination game.... Result: there is a unique Nash equilibrium where players coordinate on a in state a, in which players play a independently of the number of messages they received. Implications: strategic behavior given arbitrarily high orders of mutual knowledge is not continuous at the limit, i.e. common knowledge. Thus, common knowledge assumption is tight. Alternatively, perhaps higher-order mutual knowledge is not the correct approximation for common knowledge?

Monderer and Samet 1989, Morris 1999

Primitives: (Ω, {πi}i∈I , {Pi}i∈I) Deﬁne “i believes event E occurs with probability at least p:”

p Bi (E) := {ω ∈ Ω: πi(E|Pi(ω)) ≥ p} .

“Truth” axiom: π has full support; 1 Ki(E) = Bi (E); Fact: K(E) ∩ K(F) = K(E ∩ F), Bp(E) ∩ Bp(F) 6= Bp(E ∩ F).

Iterated belief: p p p p p p p Ii (E) = Bi (E) ∩ Bi Bj (E) ∩ Bi Bj Bk(E) ··· ; p p I (E) = ∩i∈IIi (E). Common p-belief: p p B∗(E) = ∩i∈IBi (E); p ∞ p n C (E) = ∩n=1[B∗] (E). Asymmetric case: Fix p = (p1, ··· , pn), players p-believe E if player i believes E occurs with probability at least pi for all i ∈ I; Observation: C1(E) = I1(E) = CK(E); In general: Cp(E) ⊆ Ip(E) (Morris 1999); Example.

The ﬁxed-point characterization of common p-Belief: p E is a p-evident event if: E ⊆ Bi (E) for all i ∈ I; ω ∈ Cp(G) if and only if there exists a p-evident event E such that p ω ∈ E ⊆ B∗(G). Note: Cp(G) ⊆ Cp(Cp(G)).

Suppose there is common prior, i.e. πi = πj = π for all i, j, upper bound for difference in posteriors: if ω ∈ Cp(E), then:

|π(E|Pi(ω)) − π(E|Pj(ω))| ≤ 1 − p.

(Neeman 1996)

lower bound for “common p-belief”: suppose p = (p1, ··· , pn) P satisﬁes i∈I pi < 1, then:

p 1 − mini∈I pi π(C (E)) ≥ 1 − (1 − π(E))( P ). 1 − i∈I pi (Kajii and Morris 1997)

Deﬁnition Fix a complete information game. A pure strategy Nash equilibrium a∗ is a p-dominant equilibrium if each player’s action is a best response whenever he assigns probability at least p to his opponents choosing according to a∗.

Every strict Nash equilibrium is a p-dominant equilibrium for some large p; The role of common p-belief events;

Deﬁnition A pure Nash equilibrium a∗ in the complete information game G is robust to incomplete information if for all δ < 0, there exists ε such that in every incomplete information game where the prior probability on the event “the underlying game is the complete information game G” is at least 1 − ε, a∗ is played by all players on an event with probability at least 1 − δ.

The robust notion; Strict equilibrium may not be robust; Sufﬁcient conditions for robust equilibrium: Unique correlated equilibrium; P p-dominant equilibrium with i∈I pi < 1.

Deﬁnition Global games are games of incomplete information where the type space is generated by the players each observing a noisy private signal of the underlying state.

The incomplete information interpretation versus the asymmetric information interpretation; A simple device to model higher-order uncertainties; Common knowledge assumption; the CPA. The environment itself is of interest; Equilibrium analysis; Comparative statics w.p.t. the noise technology; Limiting behavior as noise goes away.

a b a 0,0 0, θ − 1 b θ − 1, 0 θ, θ

Two players i, j; (A continuum of players, payoff: θ − 1 + l where l is the proportion of players playing b;) θ is distributed uniformly on the real line; 2 noise technology: xi = θ + εi where εi ∼ N(0, σ ), conditional on θ, signals are independent across players.

1 Equilibrium: i plays b if xi ≥ 2 , for all i; Unique; Rationalizable; Independent of the size of the noise; The Laplacian interpretation of the equilibrium; The risk-dominant equilibrium in the complete information game; The probability of players coordinating on b increases in θ; Converges as σ → 0 (trivial in this example).

A continuum of players; Binary choice: safe action (0), risky action (W¿0 if critical mass, L¡0 if not); Critical mass function α(θ) ∈ [0, 1], strictly increasing in θ in [θ, θ¯], α(θ) = 0, α(θ¯) = 1; Prob(x ≤ x∗|θ) strictly positive, continuous and decreasing in θ for any x∗; Prob(θ ≤ θ∗|x) continuous, decreasing in x for any θ∗.

The equilibrium is characterized by a pair (x∗, θ∗): Prob(x ≤ x∗|θ∗) = α(θ∗), Prob(θ ≤ θ∗|x∗)W + (1 − Prob(θ ≤ θ∗|x∗))L = 0. Iterated deletion of strictly dominated strategies: first round dominant region end points: θ, θ¯; 0 find dominant signals x0, x by driving the following to equality: 0 0 0 ¯ Prob(θ ≤ θ |x)W + (1 − Prob(θ ≤ θ |x))L T 0, θ = θ, θ. 0 find threshold states θ0, θ by driving the following to equality: 0 Prob(x ≤ ¯x|θ0) ≥ α(θ0), ¯x = x0, x . 0 second round dominant region end points: θ0, θ ; ··· ∞ n n ∞ Two monotonic sequences {xn, θn}n=0 and {x , θ }n=0; Common limit point = unique solution to the equations above.

Jing Li Topics in Foundations of Game Theory The Standard State Space Models Equilibrium Analysis The Common Prior Assumption Epistemic Conditions for Nash Equilibrium Approximate Common Knowledge Rationalizability Epistemic Foundation for Normal-Form Games Summary Epistemic Foundation for Extensive-Form Games Equilibrium Selection: Admissibility The State Space Model Revisited Nash Equilibrium

Deﬁne a normal-form game: (S, {ui}i∈I) Si: set of strategies for player i, ﬁnite; S = Πi∈ISi; Σi = ∆(Si): set of mixed strategies for i; Σ = Πi∈IΣi; ui : S → R; Expected utilities – ui :Σ → R; ∗ Nash equilibrium: σ is NE if, for all i ∈ I and all σi ∈ Σi,

∗ ∗ ∗ ui(σi , σ−i) ≥ ui(σi, σ−i).

Equilibrium in actions. Implicit learning/equilibrating process: Prior communication and self-enforcing agreement; Random matching; Focal point; The pitfall of multiple equilibrium. Equilibrium in beliefs. (the problem with mixed equilibrium) Two responses (at least): Epistemic analysis of games: What state of knowledge would keep players in Nash equilibrium? Endogenous and exogenous variables. Decision-theoretic approach: rationalizability; (Subjective approaches: Feinberg 2004a,b; Di Tillio 2004)

Primitives:(Ω, {Pi}i∈I , πi, {Si}i∈I , {fi}i∈I , {gi}i∈I) Ω: the state space; Pi: i’s possibility correspondence (information partition); πi: i’s beliefs over Ω; Si: set of strategies for i; fi :Ω → Si speciﬁes i’s action at each state; Σ gi :Ω → R speciﬁes i’s payoff function at state ω; (gi(ω):Σ → R) Belief versus knowledge; (Variants of the model: Bayesian normal-form games; state-dependent action sets; etc.) Assumption: one knows one’s own payoff function and action.

0 0 0 ω ∈ Pi(ω) ⇒ gi(ω ) = gi(ω), fi(ω ) = fi(ω).

Payoffs are u:

[u] = {ω ∈ Ω: gi(ω) = ui for all i ∈ I} .

Players play according to σ:

[σ] = {ω ∈ Ω: fi(ω) = σi for all i ∈ I} .

i is rational:    X 0 0 0  Ri = ω ∈ Ω: fi(ω) ∈ argmax πi(ω |Pi(ω))gi(ω )(si, f−i(ω )) . si∈Si 0  ω ∈Pi(ω) 

All players are rational: R = ∩i∈IRi.

1 Suppose ω ∈ ∩i∈IBi [f−i(ω)] ∩ R. Then f (ω) consists of a (pure-strategy) Nash equilibrium of the game (S, {gi}i∈I); Let I = {1, 2}. Suppose ω ∈ K∗(R ∩ [g(ω)] ∩ [φj(ω)]) where φj :Ω → ∆(Si) is deﬁned by φj(ω)(si) = πj([si]|Pj(ω)). Then φ(ω) consists of a mixed-strategy Nash equilibrium. For #(I) ≥ 3: Let π = π for all i ∈ I. Suppose i −→ −→ ω ∈ K∗(R ∩ [g(ω)]) ∩ CK([ φ (ω)]) where φ (ω) is the vector of probability distribution over Σ, one for each player. Then φi(ω) coincides for all i and it is mixed Nash;

Bottom-up approach: assuming only common knowledge of the game and rationality, what is the set of plausible outcomes?

Bernheim’s original formulation: si is rationalizable for i if si is a best response given i’s subjective belief about the opponents’ actions; and the belief only puts positive probability on j’s best responses given j’s belief, which only puts ··· , and so on. The state-space formulation: (Ω, {Pi}i∈I , {πi}i∈I , {Si}i∈I , {fi}i∈I , {ui}i∈I) ui :Σ → R: state-independent payoff function for i. Result: CK(R) ⊆ [S∞]; CK(R) = [S∞] when I = {1, 2}.

Example: The inclusion-exclusion problem (Samuelson 1992) Admissibility and forward induction; Admissibility and backward induction;

Jing Li Topics in Foundations of Game Theory The Standard State Space Models The Common Prior Assumption Introduction Approximate Common Knowledge Non-Standard Probability Systems Epistemic Foundation for Normal-Form Games Extensive-Form Rationality Epistemic Foundation for Extensive-Form Games The State Space Model Revisited The Finite Paradox

Example: the centipede game. Candidate explanations/reﬁnements: Kreps et al 1982; Aumann 1995; Reny 1993, Ben-Porath 1997, etc.. Key words: backward induction, admissibility, off-equilibrium path, counterfactuals, hypothetical reasoning; Central issue(s): rationality in extensive-form games/updating on zero-probability Samet: hypothetical knowledge analysis; Battigalli and Siniscalchi 1999; Brandenburger, Friedenberg and Keisler 2006.

Deﬁnition: (Ω, F, F 0, µ) F: a (σ)-algebra on Ω; F 0 ⊆ F \ {∅}, closed under supersets in F; µ : F × F 0 → [0, 1]; µ(E|F): probability of E conditional on event F (deﬁned even if µ(F|Ω) = 0); Countable additivity of µ; Every CPS is the limit of a sequence of full-support regular probabilities (Myerson 1986a).

Lexicographic probability system (LPS): (Ω, F, ~µ)

µi: i-th order hypothesis; The length of an LPS ~µ; LCPS: LPS with disjoint support; Updating rule: standard Bayes’ rule applied to the sequence. CLPS: a sequence of CPS; Non-Archimedean probability system (NPS): (Ω, F, µ) R(ε): minimal “hyperreal” line; µ : F → R(ε).

If Ω is finite, then CPS, LPS and NPS are equivalent. CPS and LCPS are equivalent; LPS and NPS are equivalent; Every LPS of finite length can be rephrased as an LCPS in a possibly expanded state space; The interpretation is subtle. With infinite Ω but countably additive measure: CPS and LCPS are equivalent, every LPS can be rewritten as NPS but not the converse; With non-additive measure: we do not know much.

S: set of states of nature; B: set of hypothesis (F 0), assumed to be clopen; Recursive construction: X0 = S, B0 = B; Bn−1 Xn = Xn−1 × ∆ (Xn−1) Bn−1 Bn = C ⊂ Xn : ∃ B ∈ Bn−1, C = B × ∆ (Xn−1 ; ∞ δ n T0 = Πn=0∆ (X ); T × T: the type space satisfying common certainty of coherency; The existence of a universal CPS type space: There exists a homeomorphism mapping T to ∆B(S × T); Every CPS type space can be uniquely identiﬁed with a belief-closed subset of the universal type space. Counterfactual reasoning: “conditional on event B, the opponent’s belief is...”

Deﬁne an extensive-form game (with perfect information): ((H, ), I, L, u) For any h ∈ H, 0 0 0 0 Si(h) = {s ∈ Si :(h , s(h )) h ∀h h, h ∈ Di}; For any si, H(si) = {h ∈ H : si ∈ Si(h)}; Take the set of states of nature to be the set of pure strategy proﬁles: S

Relevant hypothesis: B = {Πi∈ISi(h): h ∈ H}; State space Ω = S × Tn where T is the CPS type space; B n−1 A type: ti ∈ ∆ (S × T )

a b Rationality is deﬁned by Ri = Ri ∩ Ri where the two sets are: i is certain about his own strategy:

a Ri = {ω ∈ Ω: ti(ω)([si(ω)]|S(h)) = 1 if si ∈ Si(h)}

i plays best responses given his CPS:

b X Ri = {ω ∈ Ω: si(ω)(h) ∈ argmax ti(ω)(sj|S(h))ui((si, sj) si∈Si(h) sj∈Sj(h)

for all h ∈ H(si(ω))}

Rationality: R = R1 × R2

i is certain of E under his CPS: Ci(E) = {ω ∈ Ω: ti(ω)(E|B) = 1 for all B ∈ B} Common certainty of the opponents’ rationality (CCOR): CCOR = CCOR1 ∩ CCOR2,

CCORi = ∩n≥1Ci1 ◦ Ci2 · · · ◦ Cin (Rin+1 ), i1 = i, ik+1 6= ik

iterated belief; equivalent to common certainty of rationality under regular probabilities; with rationality and true belief, coincide again when conditional on a ﬁxed history: i.e. for any h, the event “reaching h, rationality and ‘common certainty of rationality conditional on reaching h’” is equivalent to “···

A strategy profile s is consistent with rationality if there exist some types that can justify it: i.e. there exists some ω ∈ R such that s = s(ω). (One can actually do this with finite type spaces.) Given a collection of partial histories H, the set of strategy profiles consistent with Rationality and CCOR on the set H is characterized by iterated removal of strictly dominated strategies (under CPS). Rationality and CCOR holds if and only if on the backward-induction path, (provided...) B∗ ⊆ H: the set of histories on the backward induction path. 0 R ∩ CCORH0 6= ∅ if and only H ⊆ B; ∗ h(s(ω)) ∈ B for all ω ∈ R ∩ CCORH0 .

Jing Li Topics in Foundations of Game Theory The Standard State Space Models The Common Prior Assumption Approximate Common Knowledge Epistemic Foundation for Normal-Form Games Epistemic Foundation for Extensive-Form Games The State Space Model Revisited

Conceptual Issues of the semantic approach. Self-referential problem: knowledge of the model itself? Implications of structural assumptions; Interpreting a state of the world: uncertainties in fundamentals, information, beliefs, action, etc.. The ﬁrst principle: observability. Alternative approaches. The syntactic approach: cf. Fagin, Halpern, Moses and Vardi 1995; The axiomatic approach (modal logic): Bacharach 1985, Milgrom 1981; Decision-theoretic approach: Geanakoplos 1989, Morris 1996, 1997; Implicit assumptions in the semantic approach. Example: unawareness

Bacharach (1985) Primitives: (F, K◦), where F is an algebra of events that includes the trivial and universal events, K◦ : F → F is an arbitrary set operator. Result: Take K0c, K1 − 3 as axioms on K◦. Then K◦ satisfies these axioms if and only if: 1 there exist an equivalent partitional model (Ω, P) and a knowledge operator K defined as in equation 1.1; 2 (moreover, Pm ⇔ Km for m = 1, 2, 3;) 3 with two players, the natural definition of common knowledge is equivalent to definition (1.2); 4 the information structures Pi, i = 1, 2 are common knowledge.

Milgrom (1981) Primitives: a partitional model (Ω, P) and an arbitrary set operator CK : E → E; Axioms on CK: CK(E) ⊆ E; s ∈ CK(E) ⇒ Pi(ω) ⊆ CK(E) for all i; E ⊆ F ⇒ CK(E) ⊆ CK(F); Pi(ω) ⊆ E for all i ∈ I, ω ∈ E ⇒ E = CK(E). Results: CK satisﬁes the above axioms if and only if it is deﬁned by (1.2).

Unawareness: lack of knowledge at all levels. n m U(E) ≡ ∩m=1(¬K) (E) Dekel, Lipman and Rustichini (1998a) Primitives: a state space Ω and two arbitrary set operators K, U : E → E. Axioms on U: U1: U(E) ⊆ ¬K¬K(E); U2: U(E) ⊆ UU(E); U3: KU(E) = ∅. Results: U1 − 3 + K0a ⇒ U(E) = ∅ for all E ∈ E; U1 − 3 + K0b ⇒ U(E) ⊆ ¬K(F) for all E, F ∈ E.

Jing Li Topics in Foundations of Game Theory