Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Learning coordination through new actions

Sofia B. S. D. Castro

TL;DR

This paper tackles the difficulty of achieving coordination on the payoff-dominant equilibrium in a $2\times 2$ coordination game by embedding the original game in a $2\times 3$ extension where one player has a third action. It analyzes replicator dynamics within flow-invariant subspaces and shows that, under Assumption $(A)$ and the condition $Q<0$, the $\omega$-limit of all interior initial conditions converges to the payoff-dominant outcome $(A_2,B_2)$, making it the unique attractor. The work provides concrete parameter regimes illustrated by Stag Hunt and Battle of the Sexes examples and interprets the added action as an anchor or nudging mechanism that guides learning toward the desirable equilibrium. These results offer a novel mechanism-design perspective on coordination and suggest experimental avenues to test asymmetrical action augmentation as a means to resolve coordination failures.

Abstract

We provide a novel approach to achieving a desired outcome in a coordination game: the original 2x2 game is embedded in a 2x3 game where one of the players may use a third action. For a large set of payoff values only one of the Nash equilibria of the original 2x2 game is stable under replicator dynamics. We show that this Nash equilibrium is the ω-limit of all initial conditions in the interior of the state space for the modified 2x3 game. Thus, the existence of a third action for one of the players, although not used, allows both players to coordinate on one Nash equilibrium. This Nash equilibrium is the one preferred by, at least, the player with access to the new action. This approach deals with both coordination failure (players choose the payoff-dominant Nash equilibrium, if such a Nash equilibrium exists) and miscoordination (players do not use mixed strategies).

Learning coordination through new actions

TL;DR

This paper tackles the difficulty of achieving coordination on the payoff-dominant equilibrium in a coordination game by embedding the original game in a extension where one player has a third action. It analyzes replicator dynamics within flow-invariant subspaces and shows that, under Assumption and the condition , the -limit of all interior initial conditions converges to the payoff-dominant outcome , making it the unique attractor. The work provides concrete parameter regimes illustrated by Stag Hunt and Battle of the Sexes examples and interprets the added action as an anchor or nudging mechanism that guides learning toward the desirable equilibrium. These results offer a novel mechanism-design perspective on coordination and suggest experimental avenues to test asymmetrical action augmentation as a means to resolve coordination failures.

Abstract

We provide a novel approach to achieving a desired outcome in a coordination game: the original 2x2 game is embedded in a 2x3 game where one of the players may use a third action. For a large set of payoff values only one of the Nash equilibria of the original 2x2 game is stable under replicator dynamics. We show that this Nash equilibrium is the ω-limit of all initial conditions in the interior of the state space for the modified 2x3 game. Thus, the existence of a third action for one of the players, although not used, allows both players to coordinate on one Nash equilibrium. This Nash equilibrium is the one preferred by, at least, the player with access to the new action. This approach deals with both coordination failure (players choose the payoff-dominant Nash equilibrium, if such a Nash equilibrium exists) and miscoordination (players do not use mixed strategies).
Paper Structure (17 sections, 2 theorems, 29 equations, 2 figures, 2 tables)

This paper contains 17 sections, 2 theorems, 29 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. A modification of a coordination game
  3. Dynamics in flow-invariant subspaces
  4. Learning to coordinate
  5. Assumption (A)
  6. Illustrative examples
  7. Stag Hunt:
  8. Battle of the Sexes:
  9. Concluding remarks
  10. Acknowledgements:
  11. Equilibria for the dynamics
  12. $\bullet$ when $x=0$ and $y_1y_2 \neq 0$:
  13. $\bullet$ when $x=1$ and $y_1y_2 \neq 0$:
  14. $\bullet$ when $y_1=0$ and $xy_2 \neq 0$:
  15. $\bullet$ when $y_2=0$ and $xy_1 \neq 0$:
  16. ...and 2 more sections

Key Result

Lemma 2.1

The dynamics of eq:ODE have at least 7 and at most 8 equilibria. There are no equilibria in the interior of $S$.

Figures (2)

  • Figure 1: The state space of the modified game is the wedge contained in $[0,1]^3$. The vertices are all equilibria for the dynamics of \ref{['eq:ODE']} and are labelled according to the action used by each player.
  • Figure 2: The phase diagram in $S_{y_1} \cup S_{y_2}$ under Assumption (A), depending on the sign of $b_{21}-b_{23}$.

Theorems & Definitions (4)

  • Lemma 2.1
  • proof
  • Theorem 3.1
  • proof