[Paper Review] Sequential games and nondeterministic selection functions

Joe Bolt, Jules Hedges+1 more|arXiv (Cornell University)|Nov 16, 2018

Game Theory and ApplicationsDecision Sciences5 references3 citations

TL;DR

This paper investigates nondeterministic selection functions over the finite nonempty powerset monad to model sequential games, showing they characterize iterated removal of strictly dominated strategies rather than subgame perfect equilibria. Despite the intuitive expectation, no nondeterministic selection function computes all subgame perfect Nash equilibrium plays, but they do capture a weaker rationality concept based on dominance.

ABSTRACT

This paper analyses Escardó and Oliva's generalisation of selection functions over a strong monad from a game-theoretic perspective. We focus on the case of the nondeterminism (finite nonempty powerset) monad $\mathcal{P}$. We use these nondeterministic selection functions of type $\mathcal{J}^{\mathcal{P}}_R X = (X ightarrow R) ightarrow \mathcal{P} (X)$ to study sequential games, extending previous work linking (deterministic) selection functions to game theory. Similar to deterministic selection functions, which compute a subgame perfect Nash equilibrium play of a game, we characterise those non-deterministic selection functions which have a clear game-theoretic interpretation. We show, surprisingly, no non-deterministic selection function exists which computes the set of all subgame perfect Nash equilibrium plays. Instead we show that there are selection functions corresponding to sequential versions of the iterated removal of strictly dominated strategies.

Motivation & Objective

• To analyze the game-theoretic interpretation of nondeterministic selection functions in sequential games using the finite nonempty powerset monad.
• To determine whether nondeterministic selection functions can compute all subgame perfect Nash equilibrium plays.
• To identify a game-theoretic solution concept that is precisely captured by the product of nondeterministic selection functions.
• To characterize the class of selection functions that yield consistent and meaningful strategic plays in sequential games.
• To compare the expressive power of selection functions with classical solution concepts like backward induction and iterated dominance.

Proposed method

• Uses the selection monad over the finite nonempty powerset monad $\mathcal{P}_{\mathrm{f}}$ to model nondeterministic choices in sequential games.
• Defines selection functions $\varepsilon_i: (X_i \to R) \to \mathcal{P}_{\mathrm{f}}(X_i)$ that return sets of actions not strictly dominated under a given payoff context.
• Introduces the strict dominance selection function $\varepsilon_i^s$ that returns actions whose payoff sets are not strictly dominated by any other action's payoff set.
• Applies the product operator $\bigotimes$ to combine selection functions across players, generalizing backward induction to nondeterministic settings.
• Employs the concept of upward closure and witnessing properties to analyze the consistency and rationality of selection function outputs.
• Uses game-theoretic constructions to relate the output of $\bigotimes_{i=1}^n \varepsilon_i$ to the set of plays obtainable via iterated removal of strictly dominated strategies.

Experimental results

Research questions

• RQ1 Can nondeterministic selection functions compute the full set of subgame perfect Nash equilibrium plays in sequential games?
• RQ2 What game-theoretic solution concept is captured by the product of nondeterministic selection functions over the $\mathcal{P}_{\mathrm{f}}$ monad?
• RQ3 Are there selection functions that correspond to iterated removal of strictly dominated strategies in sequential games?
• RQ4 Under what conditions is the product of witnessing and upwards closed selection functions itself witnessing?
• RQ5 How does the structure of the payoff space and semilattice order affect the outcome of selection function products?

Key findings

• No nondeterministic selection function exists that computes the set of all subgame perfect Nash equilibrium plays, contrary to initial expectations.
• The product of selection functions over the $\mathcal{P}_{\mathrm{f}}$ monad characterizes the set of plays obtainable via iterated removal of strictly dominated strategies.
• The strict dominance selection function $\varepsilon_i^s$ is both witnessing and upwards closed, providing a valid game-theoretic solution concept.
• The product $\bigotimes_{i=1}^n \varepsilon_i^s$ computes exactly the set of $\Sigma(\mathcal{G})$-plays in games where strategies are obtained by iterated removal of strictly dominated strategies.
• The product of two witnessing selection functions is not necessarily witnessing, as demonstrated by a counterexample with $X = \{0,1\}$ and $R = \mathcal{P}_{\mathrm{f}}(\mathbb{R}^2)$.
• When each $X_i$ is finite, the product $\bigotimes_{i=1}^n \varepsilon_i^s$ computes the same set of plays as the iterated removal of strictly dominated strategies in sequential games.