[Paper Review] Approximate Equilibrium and Incentivizing Social Coordination

Elliot Anshelevich, Shreyas Sekar|arXiv (Cornell University)|Apr 18, 2014

Game Theory and Applications20 references4 citations

TL;DR

This paper proposes algorithms to compute approximate equilibria in coordination games where agents have intrinsic preferences and may not coordinate efficiently due to coordination failures or non-existence of Nash equilibria. It shows that with small incentives, high-quality approximate equilibria can be achieved, and for r-supermodular games, a (r+ε)-approximate equilibrium can be computed via a one-shot best-response algorithm, ensuring stability linear in the degree of complementarity.

ABSTRACT

We study techniques to incentivize self-interested agents to form socially desirable solutions in scenarios where they benefit from mutual coordination. Towards this end, we consider coordination games where agents have different intrinsic preferences but they stand to gain if others choose the same strategy as them. For non-trivial versions of our game, stable solutions like Nash Equilibrium may not exist, or may be socially inefficient even when they do exist. This motivates us to focus on designing efficient algorithms to compute (almost) stable solutions like Approximate Equilibrium that can be realized if agents are provided some additional incentives. Our results apply in many settings like adoption of new products, project selection, and group formation, where a central authority can direct agents towards a strategy but agents may defect if they have better alternatives. We show that for any given instance, we can either compute a high quality approximate equilibrium or a near-optimal solution that can be stabilized by providing small payments to some players. We then generalize our model to encompass situations where player relationships may exhibit complementarities and present an algorithm to compute an Approximate Equilibrium whose stability factor is linear in the degree of complementarity. Our results imply that a little influence is necessary in order to ensure that selfish players coordinate and form socially efficient solutions.

Motivation & Objective

• To address coordination failures and non-existence of Nash equilibria in coordination games with intrinsic preferences.
• To design efficient algorithms that compute stable, high-welfare solutions through small monetary incentives.
• To generalize coordination games to include bounded complementarities via r-supermodular utility structures.
• To establish existence and computation of approximate equilibria with stability factors linear in the degree of complementarity.
• To provide a framework where a central authority can guide agents toward socially efficient outcomes with minimal intervention.

Proposed method

• Introduces α-approximate equilibrium as a solution concept where no player gains more than factor α by unilaterally deviating.
• Proposes a one-shot α-best-response (α-BR) algorithm that allows players to deviate only if utility improves by factor α or more.
• Uses a potential function argument to prove convergence to Nash equilibrium under the Correlated Coordination (CC) condition in hypergraph-based SCGs.
• Defines r-supermodular SCGs where utility exhibits bounded complementarities via the inequality u_i(S∪T) ≤ r(u_i(S) + u_i(T)).
• Adjusts the α-BR algorithm with α = r + ε to compute (r + ε)-approximate equilibria for r-supermodular games.
• Establishes that the stability factor is bounded by r + ε < r + 1, with both anchored and deviating players achieving comparable stability.

Experimental results

Research questions

• RQ1 Can we compute an approximate equilibrium that is both stable and socially efficient in coordination games with intrinsic preferences?
• RQ2 Under what conditions does a Nash equilibrium fail to exist in asymmetric coordination games?
• RQ3 How can a central authority provide minimal incentives to stabilize high-welfare outcomes in coordination games?
• RQ4 What is the trade-off between the degree of complementarity in utilities and the stability factor of approximate equilibria?
• RQ5 Can we achieve sublinear dependence on the complementarity parameter r in the stability factor of approximate equilibria?

Key findings

• For any instance of a social coordination game, either a high-quality approximate equilibrium can be computed, or a near-optimal solution can be stabilized with small payments to agents.
• The One-Shot α-BR algorithm with α = r + ε computes an (r + ε)-approximate Nash equilibrium for r-supermodular SCGs.
• The stability factor of the computed equilibrium is bounded by r + ε < r + 1, indicating linear dependence on the degree of complementarity.
• Under the Correlated Coordination (CC) condition, a potential function exists, ensuring convergence of best-response dynamics to a Nash equilibrium.
• The paper establishes that even in games without pure Nash equilibria, approximate equilibria with bounded stability factors can be computed efficiently.
• The results imply that minimal incentives—proportional to a fraction (α−1) of utility—can stabilize socially efficient outcomes.