Imperfect-Recall Games: Equilibrium Concepts and Their Complexity

TL;DR

In the framework of extensive-form games with imperfect recall, the computational complexities of finding equilibria in multiplayer settings are analyzed across three different solution concepts: Nash, multiselves based on evidential decision theory (EDT), and multiselves based on causal decision theory (CDT).

Abstract

We investigate optimal decision making under imperfect recall, that is, when an agent forgets information it once held before. An example is the absentminded driver game, as well as team games in which the members have limited communication capabilities. In the framework of extensive-form games with imperfect recall, we analyze the computational complexities of finding equilibria in multiplayer settings across three different solution concepts: Nash, multiselves based on evidential decision theory (EDT), and multiselves based on causal decision theory (CDT). We are interested in both exact and approximate solution computation. As special cases, we consider (1) single-player games, (2) two-player zero-sum games and relationships to maximin values, and (3) games without exogenous stochasticity (chance nodes). We relate these problems to the complexity classes P, PPAD, PLS, $Σ_2^P$ , $\exists$R, and $\exists \forall$R.

Figures (10)

• Figure 1: Games with imperfect recall. P1's ($\color{p1color}\blacktriangle$) utility payoffs are labeled on each terminal node. If P2 ($\color{p2color}\blacktriangledown$) is present, the game is zero sum. Infosets are joined by dotted lines.
• Figure 2: Game construction used to prove hardness of deciding equilibrium existence. We use boxes for chance nodes, at which chance plays uniformly at random. $\Gamma$ is a placeholder game. G is a game with no equilibrium; \ref{['sec:decide NE exists']} for example uses \ref{['fig:forgetting kicker game']}.
• Figure 3: A single-player game with imperfect recall where miscoordinating actions with yourself is punished most.
• Figure 4: Differences of the ex-ante and de-se utility perspective explained on a perfect-recall variant of \ref{['fig:coord problem']}. Again, the only optimal strategy takes the path $r_1$ -- $r_3$. But what action can you choose at $h_2$?
• Figure 5: A variant of \ref{['fig:forgetting kicker game']} where P1 has one single infoset with absentmindedness. It is parametrized by the payoff $\lambda \in \mathbb{R}$ from P1 shooting left and P2 blocking left.

Theorems & Definitions (114)

• Definition 1
• Remark
• Definition 2
• Proposition 3: ?, ?
• Proposition 4: ?, ?; ?, ?
• Definition 5
• Lemma 6: ?, ?
• Proposition 7
• Theorem 1
• Theorem 2