Game Theory with Simulation of Other Players

TL;DR

This paper formally defines games in which one player can simulate another at a cost, and derive some basic properties of such games, and proves a number of results, including that introducing simulation into generic-payoff normal-form games makes them easier to solve.

Abstract

Game-theoretic interactions with AI agents could differ from traditional human-human interactions in various ways. One such difference is that it may be possible to simulate an AI agent (for example because its source code is known), which allows others to accurately predict the agent's actions. This could lower the bar for trust and cooperation. In this paper, we formalize games in which one player can simulate another at a cost. We first derive some basic properties of such games and then prove a number of results for them, including: (1) introducing simulation into generic-payoff normal-form games makes them easier to solve; (2) if the only obstacle to cooperation is a lack of trust in the possibly-simulated agent, simulation enables equilibria that improve the outcome for both agents; and however (3) there are settings where introducing simulation results in strictly worse outcomes for both players.

Figures (9)

• Figure 1: The underlying trust game $\textnormal{TG}$ (left) and the corresponding simulation game $\textnormal{TG}_{\textnormal{sim}}$ (right).
• Figure 2: Top left: The normal-form representation of the trust game from Figure \ref{['fig:trust-game-efg']}, before and after adding simulation. Bottom: The extremal equilibria of $\textnormal{TG}_{\textnormal{sim}}^\texttt{c}$. The non-extremal NE are precisely the convex combinations of the last two columns. Top right: The cooperation probability and utilities under each of these NE. The non-extremal NE are light red, the dashed lines illustrate the NE trajectories from Proposition \ref{['prop:piecewise_constant_linear']}. Note that all the red NE (i.e., with $\pi_1(\textnormal{WO}) = 1$) yield $u_1 = u_2 = 0$.
• Figure 3: Left: Commitment game, where the row player prefers to not be able to simulate. For details, see Example \ref{['ex:commitment-game-continued']}. Right: A variant of Trust Game with multiple simulation NE.
• Figure 4: Cafés in Paris: Alice and Bob want to meet but they need to coordinate on which café to go to. We assume that $x_i, y_i > 0$ for every $i$. The actual game has $n \in \mathbb{N}$ actions.
• Figure 5: The game $\widetilde{{\mathcal{G}}}$ which shows that for any ${\mathcal{G}}$, there is a similarly-sized simulation game that is as hard to solve as ${\mathcal{G}}$.

Theorems & Definitions (78)

• Definition 1: Generic games
• Definition 2: No best-response utility tiebreaking
• Definition 4: Simulation game
• Lemma 4
• Lemma 4
• Proposition 4: Equilibria for extreme simulation costs
• Definition 5: Value of information of simulation
• Lemma 5
• Lemma 5: $\VoI$ is equal to simulation cost
• Proposition 5: Trajectories of simulation NE are piecewise constant/linear