Coordination game
Games where players want to match each other, and the only problem is which convention to pick. Pure coordination, the better-and-worse variant, and battle of the sexes.
Coordination game
Players want to do the same thing as each other. Nobody gains by breaking the match, so there is no temptation. The problem is picking which match.
Pure coordination
Which side of the road to drive on. Payoffs (row, column):
| Column A | Column B | |
|---|---|---|
| Row A | 1, 1 | 0, 0 |
| Row B | 0, 0 | 1, 1 |
Two pure equilibria, both A or both B, and a mixed one where each picks at random. No greed and no fear in the sense of Greed and fear; it sits off the greed/fear map as the easy baseline.
If one convention is better (Osborne and Rubinstein's two concerts, 2, 2 against 1, 1), both are still equilibria.
a steady state in which the outcome is the inferior equilibrium
— Martin J. Osborne and Ariel Rubinstein, A Course in Game Theory (1994), p. 16, on what Nash equilibrium does not rule out
Reading this as the Stag hunt with the risk taken away is this wiki's gloss, not Osborne and Rubinstein's.
Battle of the sexes
Both want to match, but each prefers a different match. Osborne and Rubinstein call it BoS, "Bach or Stravinsky" (p. 15).
| Column Bach | Column Stravinsky | |
|---|---|---|
| Row Bach | 2, 1 | 0, 0 |
| Row Stravinsky | 0, 0 | 1, 2 |
Two pure equilibria. The mixed equilibrium gives each player an expected 2/3, worse than either pure one (pp. 34–35; p. 46). Relabel the moves "insist on mine" and "give way" and it has the shape of Chicken (game): a fight over who yields. The difference is that in chicken mutual insistence is the disaster, while in battle of the sexes both mismatches are bad. The course groups it with chicken.
Fixes
- A focal point, convention or signal (Schelling 1960 [?]; Lewis 1969 [?]).
- Taking turns. With a public random signal that both see, there is a correlated equilibrium that alternates between the two Bach-or-Stravinsky equilibria (Osborne and Rubinstein p. 44).
Cases
- Driving on one side of the road; technical standards. The Universal Postal Union is a standards case.
- Standards wars, where each side prefers its own format, are battle of the sexes.
- Synchronized planting in Balinese subak irrigation (Lansing [?]) is coordination; the upstream–downstream water split in the same system is a taking problem.
Repeated play turns a Prisoner's Dilemma into a coordination problem among many equilibria (Osborne and Rubinstein p. 134).
