This chapter supplies the rigorous core of non-cooperative theory: why equilibria exist, and how to discipline the multiplicity of equilibria in dynamic games.
Fixed-point theorems
Brouwer’s fixed-point theorembrouwer Every continuous function \(f : K \to K\) from a nonempty compact convex set \(K \subset \R^m\) to itself has a fixed point: \(f(x^*) = x^*\).
Kakutani’s fixed-point theorem (1941)kakutani Let \(K \subset \R^m\) be nonempty, compact, convex, and let \(F : K \rightrightarrows K\) be a correspondence such that (i) \(F(x)\) is nonempty and convex for every \(x\), and (ii) \(F\) has a closed graph. Then \(F\) has a fixed point: \(x^* \in F(x^*)\).
Nash’s theorem
Nash (1950)nashexist Every finite game \(G = \langle N, (A_i), (u_i) \rangle\) has at least one Nash equilibrium in mixed strategies.
Proof. Let \(\Sigma = \prod_i \Delta(A_i)\), a nonempty compact convex subset of a Euclidean space. Define the best-response correspondence \(B(\sigma) = \prod_i \BR_i(\sigma_{-i})\), where \(\BR_i(\sigma_{-i}) = \argmax_{\tau_i \in \Delta(A_i)} u_i(\tau_i, \sigma_{-i})\). Each \(\BR_i(\sigma_{-i})\) is nonempty (a continuous function is maximized on a compact set) and convex (the maximizers of a linear function on a simplex form a face, which is convex). The graph of \(B\) is closed because \(u_i\) is continuous in \(\sigma\) (Berge’s maximum theorem, or a direct limit argument). By Kakutani’s theorem, \(B\) has a fixed point \(\sigma^*\), which by construction satisfies \(\sigma_i^* \in \BR_i(\sigma^*_{-i})\) for all \(i\): a Nash equilibrium. ◻
Nash (1951) gave a self-contained proof using Brouwer directly: define the “gain” \(g_{i a} (\sigma) = \max\{0,\, u_i(a, \sigma_{-i}) - u_i(\sigma)\}\) and the continuous map \[T_{ia}(\sigma) = \frac{\sigma_i(a) + g_{ia}(\sigma)}{1 + \sum_{a'} g_{ia'}(\sigma)} .\] Fixed points of \(T\) are exactly the Nash equilibria (if \(\sigma\) is not an equilibrium, probability flows toward better replies and \(\sigma\) moves). This map is also the starting point for computational approaches—and for understanding why computing the fixed point is hard (Chapter [ch:algorithmic]).
Games with continuous action spaces (Debreu–Glicksberg–Fan 1952)dgf If each \(A_i\) is a nonempty compact convex subset of \(\R^m\), and each \(u_i\) is continuous in \(a\) and quasi-concave in own action \(a_i\), then a pure-strategy Nash equilibrium exists.
This covers Cournot, Bertrand with differentiated goods, public-good games, etc. Glicksberg (1952) further extends Nash’s theorem to continuous games (compact metric action spaces, continuous payoffs) with mixed strategies. Discontinuous games (all-pay auctions, Bertrand) need more delicate arguments (Reny’s better-reply security, 1999).
Trembling-hand perfection
Nash equilibrium can rely on weakly dominated actions and incredible behavior at unreached information sets. Selten’s idea: robustness to small mistakes.
Trembling-hand perfect equilibrium (Selten 1975)thp A mixed profile \(\sigma^*\) of a finite normal-form game is (trembling-hand) perfect if there is a sequence of totally mixed profiles \(\sigma^k \to \sigma^*\) (every action played with positive probability) such that each \(\sigma_i^*\) is a best response to \(\sigma_{-i}^k\) for every \(k\).
Properties of perfectionthpprops Every finite game has a perfect equilibrium. Every perfect equilibrium is Nash, and assigns zero probability to weakly dominated actions. In two-player games, an equilibrium is perfect iff both strategies are undominated.
Ruling out dominated equilibriathpexample
| \(L\) | \(R\) | |
|---|---|---|
| \(T\) | \(1,1\) | \(0,0\) |
| \(B\) | \(0,0\) | \(0,0\) |
Both \((T,L)\) and \((B,R)\) are Nash equilibria, but \((B,R)\) uses weakly dominated actions on both sides: any tremble by the opponent makes \(T\) (resp. \(L\)) strictly better. Only \((T, L)\) is perfect.
Sequential equilibrium
For extensive-form games, the workhorse refinement pairs strategies with beliefs, like PBE, but disciplines off-path beliefs by a limiting consistency requirement.
Sequential equilibrium (Kreps–Wilson 1982)seqeq An assessment \((\sigma, \mu)\)—a behavior strategy profile plus a belief system —is a sequential equilibrium if:
Sequential rationality: at every information set, the mover’s continuation strategy is optimal given \(\mu\) and \(\sigma\);
Consistency: \((\sigma, \mu) = \lim_k (\sigma^k, \mu^k)\) for some sequence of totally mixed \(\sigma^k\) with \(\mu^k\) derived from \(\sigma^k\) by Bayes’ rule everywhere.
Consistency says off-path beliefs must be explicable as limits of small correlated trembles—ruling out arbitrary belief threats that plain PBE permits. Every finite extensive-form game with perfect recall has a sequential equilibrium; every sequential equilibrium is subgame perfect; trembling-hand perfection in the agent normal form implies sequentiality.
For finite extensive-form games with perfect recall: \[\text{THP (agent form)} \subseteq \text{sequential} \subseteq \text{PBE} \subseteq \text{subgame perfect} \subseteq \text{Nash}.\] Further refinements—properness (Myerson), stability (Kohlberg–Mertens), forward induction, the intuitive criterion (Theorem [thm:intuitive])—address signaling and forward-looking inferences. The refinements program of the 1980s never produced a single universally accepted concept, but its ideas (credibility, consistency of beliefs, forward induction) permanently shaped applied modeling.
Epistemic foundations
What assumptions on knowledge and belief justify each solution concept? A sample of landmark results from epistemic game theory:
Rationalizability: common knowledge of rationality characterizes (correlated) rationalizability, i.e. IESDS survivors (Theorem [thm:ratiesds]).
Nash equilibrium (Aumann–Brandenburger 1995): in two-player games, mutual knowledge of rationality and of the players’ conjectures about each other implies the conjectures form a Nash equilibrium—common knowledge is not needed, but knowing the opponent’s belief is a strong requirement.
Correlated equilibrium (Aumann 1987): if players share a common prior and are “Bayes rational” given their information, their behavior is a correlated equilibrium. This gives correlated equilibrium a cleaner epistemic pedigree than Nash.
Agreeing to disagree (Aumann 1976): with a common prior, if two players’ posteriors about an event are common knowledge, they must be equal—rational agents cannot agree to disagree. Corollary (no-trade theorems, Milgrom–Stokey 1982): rational risk-averse traders with a common prior cannot trade on differences of information alone—if someone wants to bet with you, that itself is information (compare the winner’s curse, Example [exa:winnerscurse]).
Exercises
Existence via Kakutani, hands-onkakutaniexercise Consider a two-player game where each player picks \(x_i \in [0,1]\) and \(u_i = -(x_i - \tfrac12 x_j)^2\). Compute best responses, verify the conditions of Theorem [thm:dgf], and find the unique pure equilibrium.
Perfection in the normal formperfectexercise
| \(L\) | \(C\) | \(R\) | |
|---|---|---|---|
| \(T\) | \(2,2\) | \(1,1\) | \(0,0\) |
| \(M\) | \(1,1\) | \(1,1\) | \(1,1\) |
| \(B\) | \(0,0\) | \(1,1\) | \(2,2\) |
Find all pure Nash equilibria and determine which are trembling-hand perfect.
Sequential vs. PBEseqvspbe Construct (or recall from the literature) a three-player extensive-form game with a PBE whose off-path beliefs cannot arise as limits of trembles, so the assessment is not sequential. Explain exactly where consistency bites.
No-tradenotrade Two risk-neutral agents with a common prior each receive a private signal about an asset’s value and are offered the chance to bet against each other; betting is voluntary. Prove that in any equilibrium, the probability of a bet actually occurring is zero. Where exactly does the common prior enter, and what happens without it?