- Content map: SMU H3 Game Theory Map
Setup
Definition:
Common-Knowledge Coordination Game
- Players: Commander 1 and Commander 2.
- Strategies: Each commander chooses Attack or Wait.
- Rules: Players choose after finite confirmations; attacking is safe only with common knowledge of coordination.
Payoff Matrix
| Attack | Wait | |
|---|---|---|
| Attack | 1, 1 | -1, -1 |
| Wait | -1, -1 | 0, 0 |
Common Knowledge
- Common knowledge requires that all players know a fact, know that others know it, know that others know they know it, and so on ad infinitum.
- In this game, to safely choose Attack, each commander must have common knowledge that the other will attack.
- Unreliable messages prevent finite confirmation chains from establishing common knowledge.
Derivation (Logic of Confirmations)
- Suppose Commander A sends Attack and receives B’s confirmation.
- A still doubts if B received A’s acknowledgement of B’s confirmation, creating an infinite regress of uncertainty.
- No finite sequence of messages can eliminate the possibility of failure at some level, so doubt persists.
- Thus, rational players must choose Wait to avoid the risk of unilateral attack.
Nash Equilibrium
Result:
No finite number of confirmations can guarantee successful coordination.
Insights
Insight:
Coordination is fragile when it requires common knowledge.
“Everyone knows” is not enough, as players must know that others know, and so on.