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Smart contracts and reaction-function games

Jens Gudmundsson, Jens Leth Hougaard

TL;DR

This paper studies blockchain-enabled reaction-function games, where players commit to conditional actions via smart contracts as a one-shot alternative to repeated-game triggers. It develops a formal fixed-point framework, proves existence of reaction-function equilibria, and characterizes which outcomes can be supported, including a maxmin-based Folk Theorem analogue for two players. It then introduces safe play, norm-proofness, and payoff-consistency to refine equilibria, showing how these refinements limit multiplicity while preserving the potential for Pareto improvements. The authors apply the theory to symmetric investment games, deriving concrete, safe reaction rules for weakest-link and public-good settings and outlining a practical implementation with escrowed deposits and automated transfers, thereby illustrating how blockchain commitment can overcome trust and free-riding barriers in coordination problems.

Abstract

Blockchain-based smart contracts offer a new take on credible commitment, where players can commit to actions in reaction to actions of others. Such reaction-function games extend on strategic games with players choosing reaction functions instead of strategies. We formalize a solution concept in terms of fixed points for such games, akin to Nash equilibrium, and prove equilibrium existence. Reaction functions can mimic "trigger" strategies from folk theorems on infinitely repeated games -- but now in a one-shot setting -- for instance to support Pareto-improvements on Nash equilibrium outcomes. In some games, this can even be done through risk-free, safe reaction functions. We apply our theoretical framework to symmetric investment games, which includes two prominent classes of games, namely weakest-link and public-good games. In both cases, we highlight a particular safe and optimal reaction function. In this way, our findings highlight how blockchain-based commitment can help overcome trust and free-riding barriers.

Smart contracts and reaction-function games

TL;DR

This paper studies blockchain-enabled reaction-function games, where players commit to conditional actions via smart contracts as a one-shot alternative to repeated-game triggers. It develops a formal fixed-point framework, proves existence of reaction-function equilibria, and characterizes which outcomes can be supported, including a maxmin-based Folk Theorem analogue for two players. It then introduces safe play, norm-proofness, and payoff-consistency to refine equilibria, showing how these refinements limit multiplicity while preserving the potential for Pareto improvements. The authors apply the theory to symmetric investment games, deriving concrete, safe reaction rules for weakest-link and public-good settings and outlining a practical implementation with escrowed deposits and automated transfers, thereby illustrating how blockchain commitment can overcome trust and free-riding barriers in coordination problems.

Abstract

Blockchain-based smart contracts offer a new take on credible commitment, where players can commit to actions in reaction to actions of others. Such reaction-function games extend on strategic games with players choosing reaction functions instead of strategies. We formalize a solution concept in terms of fixed points for such games, akin to Nash equilibrium, and prove equilibrium existence. Reaction functions can mimic "trigger" strategies from folk theorems on infinitely repeated games -- but now in a one-shot setting -- for instance to support Pareto-improvements on Nash equilibrium outcomes. In some games, this can even be done through risk-free, safe reaction functions. We apply our theoretical framework to symmetric investment games, which includes two prominent classes of games, namely weakest-link and public-good games. In both cases, we highlight a particular safe and optimal reaction function. In this way, our findings highlight how blockchain-based commitment can help overcome trust and free-riding barriers.

Paper Structure

This paper contains 15 sections, 15 theorems, 30 equations, 2 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Games and definitions
  3. Strategic games
  4. Reaction functions
  5. Fixed points in reaction-function games
  6. Reaction-function equilibrium
  7. What outcomes can be supported?
  8. Equilibrium existence
  9. Maxmin payoff bound for two-player games
  10. More players, more possibilities for support
  11. "Playing it safe" with reaction functions
  12. Symmetric investment games
  13. Weakest-link games
  14. Public-good games
  15. Concluding remarks

Key Result

Theorem 3.1

In every reaction-function game, there is a reaction-function equilibrium.

Figures (2)

  • Figure 1: Caption
  • Figure 2: The dashed line is to match the average contribution of the others. Open circles is $R^*_i$, to round down the average. The solid curve shows simulation averages and filled dots are simulation median values. Simulation medians match $R^*_i$ whenever the dot is encircled.

Theorems & Definitions (32)

  • Remark 1: Sufficient condition for unique fixed point
  • Definition 1
  • Definition 2
  • Example 1: Deviating from best-reply reactions
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Definition 3
  • Theorem 4.1
  • Example 2: No safe reaction-function equilibrium
  • ...and 22 more