Assurance game
The assurance game is a coordination game with two Nash equilibria that is often confused with the stag hunt.
Payoff Matrix
| P2: Cooperate | P2: Defect | |
|---|---|---|
| P1: Cooperate | II | III |
| P1: Defect | I | IV |
- P1 prefers: II > I > IV > III
- P2 prefers: II > III > IV > I
There are two Nash equilibrium strategies:
- Box II — both cooperate. The better outcome for both players.
- Box IV — both defect. Stable but worse than mutual cooperation.
In the stag hunt, defecting on a cooperating player (boxes I or III) is no better than mutual defection. So switching to cooperation carries limited downside — if the other player doesn’t follow, you’re no worse off than before.
In the Assurance Game, the situation is different:
- P1 prefers I > IV — defecting when P2 cooperates is better than mutual defection.
- P2 prefers III > IV — defecting when P1 cooperates is better than mutual defection.
This means that if you decide to switch from box IV to box II by cooperating, and the other player doesn’t follow, you end up in box III — the worst possible outcome. This risk reduces each player’s confidence that the other will cooperate, making the move to box II harder to justify even when both players would prefer it.
Example: A coupon war between competitors (e.g., Papa John’s vs. Domino’s). Both firms would prefer mutual restraint (II) to a costly coupon war (IV). But if one firm holds back while the other runs coupons, the restrained firm loses badly (III). This fear of being the only one to cooperate keeps both firms stuck in IV.