Extensive Form Games

Abstract

After a slight reformulation of the sequential equilibrium concept, we display the set of sequential equilibria of a finite extensive form game as the set of fixed points of a upper hemicontinuous contractible valued best response correspondence. A simple signalling game illustrates the main concepts, after which we lay out the formalism of extensive form games. A behavior strategy for an agent is an assignment of a probability distribution over the available actions at each of the agent’s information sets. A belief is an assignment to each information set of a probability distribution over the information set. When play is governed by an interior behavior strategy profile, each node occurs with positive probability, so there is an induced belief given by taking conditional probabilities. A consistent assessment is a behavior strategy profile-belief pair that is a limit of a sequence of interior behavior strategy profile-induced belief pairs. A sequential equilibrium is a consistent assessment in which behavior at each information set maximizes expected utility, as computed from the belief at the information set and the strategy profile. A consistent conditional system includes additional conditional probabilities beyond those in a consistent assessment. We show that the best response correspondence for consistent conditional systems is contractible valued; in a sense, including additional information “unfolds” the best response correspondence for consistent assessments. Existence proofs for previously known refinements of the sequential equilibrium concept are obtained by showing that certain subcorrespondences are upper hemicontinuous and contractible valued.

Exercises

16.1

Let $G = (T, \prec , H, (A_h)_{h \in H}, I, \iota , \rho , u)$ be an extensive form game. Fixing $t \in T$ and a mixed strategy profile $\sigma$, write the probability that t occurs, when play is governed by $\sigma$, as the probability of the initial predecessor of t times the product over agents i of the probability that i plays a pure strategy that allows t to occur. Write the probability that i plays a pure strategy that allows t as the product, over predecessors $t'$ of t in information sets at which i chooses the action, of the probability, conditional on allowing $t'$, of choosing the action that leads toward t. Argue that if the game satisfies perfect recall, then this conditional probability is the same as the probability of choosing the action that leads to t conditional on allowing $\eta (t')$ to occur. Prove Kuhn’s theorem, as asserted in Sect. 16.3.

16.2

Let $G = (T, \prec , H, (A_h)_{h \in H}, I, \iota , \rho , u)$ be an extensive form game with perfect recall, and fix $i \in I$. The personal decision tree of i is the pair $(T_i,\prec _i)$ with the following description. The set of nodes is $T_i := \{o_i\} \cup H_i \cup A_i \cup Z$ where $o_i$ is an artificial initial node. The personal precedence relation $\prec _i$ has the following cumbersome but natural description:

• $o_i$ precedes all other elements of $T_i$;

• $h \prec _i h'$ if ${\overline{P}}(x') \cap h \ne \emptyset$ for some (hence all, by perfect recall) $x' \in h'$;

• $h \prec _i a$ if $h = h_a$ or $h \prec _i h_a$;

• $a \prec _i h$ if $a \in \alpha (\underline{P}(x))$ for some (hence all, by perfect recall) $x \in h$;

• $a \prec _i a'$ if $a \prec _i h_{a'}$,

• $z \prec _i h$ if ${\overline{P}}(z) \cap h \ne \emptyset$;

• $z \prec _i a$ if $a \in \alpha (\underline{P}(z))$.

Let $(\mu , \pi )$ be an interior consistent assessment.

• (a) Show that the expected payoff of i, conditional on any $t_i \in T_i$, is the sum, over immediate successors of $t_i$ in $(T_i,\prec _i)$, of the probability of transitioning to the successor times that expected payoff conditional on the successor.

• (b) Show that the probability of transitioning from $a \in A_i$ to an immediate successor in $(T_i,\prec _i)$ does not depend on $\pi _i$.

• (c) Prove that if a (not necessarily interior) consistent assessment is myopically rational, then it is sequentially rational.

16.3

(This continues from the last exercise.)

• (a) Let $(\mu ,\pi )$ be an interior consistent assessment. Using i’s personal decision tree, show that the set of $u_i \in \mathbb {R}^Z$ such that ${\mathbf {E}}^{\mu ,\pi |a}(u_i|h) = {\mathbf {E}}^{\mu ,\pi |a'}(u_i|h)$ for all $h \in H_i$ and $a, a' \in A_h$ is a linear subspace of dimension $|Z| + |H_i| - |A_i|$.

• (b) Let M be the set of triples $(u,\mu ,\pi )$ such that $(\mu ,\pi )$ is an interior consistent assessment that is a sequential equilibrium for the payoff profile $u \in (\mathbb {R}^Z)^I$. Prove that M is an $(|I| \times |Z|)$-dimensional $C^\infty$ manifold.

• (c) Applying Sard’s theorem to the projection $(u,\mu ,\pi ) \mapsto u$ of M onto U, prove that for generic payoff profiles u there are finitely many interior sequential equilibria.

• (d) The path of a behavior strategy is the induced distribution on Z: the path assigns probability ${\mathbf {P}}^\pi (z)$ to each z. Prove that a sequential equilibrium $(\mu ,\pi )$ “projects” to an interior sequential equilibrium of the truncated extensive game obtained by eliminating all parts of the extensive form that have zero probability under the path of $\pi$.

• (e) Prove that for generic payoff profiles u there are finitely many distributions on Z that are paths of sequential equilibria.

16.4

Holding the combinatoric data $(T,\prec ,H,(A_h)_{h \in H}, I,\iota )$ of an extensive game of perfect recall fixed, prove that the correspondence that takes each initial assessment-payoff pair $(\rho , u)$ to the set of sequential equilibria of the extensive game $(T,\prec ,H,(A_h)_{h \in H},I,\iota ,\rho , u)$ is upper hemicontinuous.

16.5

(McLennan (1985)) Find the set of sequential equilibria of the game below. Which have beliefs that are first order justifiable? Which have beliefs that are second order justifiable?

16.6

Sketch two different proofs that our given extensive form game has a sequential equilibrium with second order justifiable beliefs.

• (a) Give a sequence of trembles $\varepsilon ^r$ for the agent normal form such that if, for each r, $\pi ^r$ is a $\varepsilon ^r$-perfect equilibrium and $(\mu ^{\pi ^r},\pi ^r) \rightarrow (\mu ,\pi )$, then $\mu$ is second order justifiable.

• (b) As in Lemma 16.23, define an appropriate set of orderings S, and show that $K_S$ is star-shaped.

16.7

Devise a signalling game with two messages, one of which has only one response while the other has three responses, such that all sequential equilibria satisfy the intuitive criterion, but some equilibrium does not survive the iterative elimination procedure described in Sect. 16.9.