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Choosing What Game to Play without Selecting Equilibria: Inferring Safe (Pareto) Improvements in Binary Constraint Structures

Caspar Oesterheld, Vincent Conitzer

Abstract

We consider a setting in which a principal gets to choose which game from some given set is played by a group of agents. The principal would like to choose a game that favors one of the players, the social preferences of the players, or the principal's own preferences. Unfortunately, given the potential multiplicity of equilibria, it is conceptually unclear how to tell which of even any two games is better. Oesterheld et al. (2022) propose that we use assumptions about outcome correspondence -- i.e., about how the outcomes of different games relate -- to allow comparisons in some cases. For example, it seems reasonable to assume that isomorphic games are played isomorphically. From such assumptions we can sometimes deduce that the outcome of one game G' is guaranteed to be better than the outcome of another game G, even if we do not have beliefs about how each of G and G' will be played individually. Following Oesterheld et al., we then call G' a safe improvement on G. In this paper, we study how to derive safe improvement relations. We first show that if we are given a set of games and arbitrary assumptions about outcome correspondence between these games, deriving safe improvement relations is co-NP-complete. We then study the (in)completeness of a natural set of inference rules for outcome correspondence. We show that in general the inference rules are incomplete. However, we also show that under natural, generally applicable assumptions about outcome correspondence the rules are complete.

Choosing What Game to Play without Selecting Equilibria: Inferring Safe (Pareto) Improvements in Binary Constraint Structures

Abstract

We consider a setting in which a principal gets to choose which game from some given set is played by a group of agents. The principal would like to choose a game that favors one of the players, the social preferences of the players, or the principal's own preferences. Unfortunately, given the potential multiplicity of equilibria, it is conceptually unclear how to tell which of even any two games is better. Oesterheld et al. (2022) propose that we use assumptions about outcome correspondence -- i.e., about how the outcomes of different games relate -- to allow comparisons in some cases. For example, it seems reasonable to assume that isomorphic games are played isomorphically. From such assumptions we can sometimes deduce that the outcome of one game G' is guaranteed to be better than the outcome of another game G, even if we do not have beliefs about how each of G and G' will be played individually. Following Oesterheld et al., we then call G' a safe improvement on G. In this paper, we study how to derive safe improvement relations. We first show that if we are given a set of games and arbitrary assumptions about outcome correspondence between these games, deriving safe improvement relations is co-NP-complete. We then study the (in)completeness of a natural set of inference rules for outcome correspondence. We show that in general the inference rules are incomplete. However, we also show that under natural, generally applicable assumptions about outcome correspondence the rules are complete.

Paper Structure

This paper contains 26 sections, 20 theorems, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Safe (Pareto) improvements and outcome correspondence
  4. Complexity results under general outcome correspondence assumptions
  5. Conditions for completeness of rule-based inference
  6. Further outcome correspondence assumptions
  7. Inference is complete if the outcome correspondence assumptions are max-closed
  8. Generalizing completeness to infinite binary constraint structures
  9. Related work
  10. Completeness and incompleteness of inference as per \ref{['lemma:basic-results']} and computational complexity under natural generalizations of max-closedness
  11. Refutation completeness of inference as per \ref{['lemma:basic-results']}
  12. Incompleteness of \ref{['lemma:basic-results']} under join-closedness and similar conditions
  13. Inference under join-closedness can be solved in polynomial time
  14. Relation between the formalism of Oesterheld et al. oesterheld2022safe and ours
  15. Incompleteness of inference as per \ref{['lemma:basic-results']}
  16. ...and 11 more sections

Key Result

Lemma 0

Let $X$, $Y$ and $Z$ be variables of a BCS with domains $D_X$, $D_Y$ and $D_Z$, and $\Phi,\Xi \colon D_X\multimap D_Y$, $\Psi\colon D_Y \multimap D_Z$. Then:

Figures (5)

  • Figure 1: An example of three normal-form games with plausible outcome correspondences between them illustrated by lines: If playing $\Gamma_b$ would result in $(D,D)$, then playing $\Gamma_c$ would result in $(F,F)$, etc.
  • Figure 2: A type of outcome correspondence to which we can reduce binary constraint satisfaction problems without producing wildly implausible outcome correspondences.
  • Figure 3: Example to illustrate \ref{['assumption:decreasing-risk']}. From left to right, $a_H$ only becomes more attractive for both players.
  • Figure 4: An illustration of the kinds of OCs that are not allowed under max closedness. Arrows indicate orderings ($\geq_X$, $\geq_Y$); lines indicate OCs.
  • Figure 5: Semilattices for the example for the proof of \ref{['prop:incompleteness-inference-join-semi-lattice']}

Theorems & Definitions (37)

  • Lemma 0
  • Definition 1
  • Proposition 1
  • Theorem 2
  • Definition 2: jeavons1995tractable
  • Theorem 3
  • Proposition 3
  • Corollary 3
  • Theorem 4
  • Corollary 5
  • ...and 27 more