An arms race, a team project, a couple picking a night out, two cars in a game of nerve, a goalkeeper facing a penalty. The same two-by-two shape, with the payoffs nudged, describes all of them, and each has a different kind of answer.
Classic 2x2 games
The Prisoner’s Dilemma is one animal in a small zoo. Change the four payoffs and you get coordination games, games of nerve, and games with no stable move at all. Learning the family is worth more than memorizing any one game, because the structure tells you what to look for.
In the Prisoner's Dilemma, both players defect because defecting is...
Both prisoners would gain if they cooperated together. What does that show about mutual defection?
Predict, then play
Here is one 2x2 you can retune. Every cell holds two payoffs: yours on top, theirs below. Load each classic game in turn, then nudge the payoffs and watch the family label and the equilibria (gold cells) shift as incentives cross their thresholds.
Why did that happen?
Take a guess, then reveal the reason
The four numbers are the whole story. Where the temptation to deviate sits, and whether players want to match or to differ, decides everything: whether there is a dominant move, how many stable cells there are, and whether the stable cells make players agree or fight over which one to reach. Nudge one payoff past a threshold and a coordination game can turn into a dilemma. The families are regions in payoff space, not separate universes.
The math
Five games map the territory.
Prisoner’s Dilemma (R3, S0, T5, P1): defecting dominates, so the unique Nash is mutual defection, worse for both than cooperating. Stag Hunt ((4,4)/(0,3)/(3,0)/(3,3)): two Nash on the diagonal. All-Stag pays more (payoff-dominant), while Hare guarantees and Stag can pay against Hare. For these payoffs, the basin-of-attraction calculation also selects all-Hare as risk-dominant. Battle of the Sexes ((2,1)/(0,0)/(0,0)/(1,2)): two Nash, but each player prefers a different one. Coordination with conflict. Chicken ((6,6)/(2,7)/(7,2)/(0,0)): two Nash off the diagonal. Each wants to do the opposite of the other, and the crash is worst for both. Matching Pennies ((1,-1)/(-1,1)/(-1,1)/(1,-1)): one wants to match, the other to differ. No pure Nash exists.
Three lessons hide in this list. Not every dilemma is a Prisoner’s Dilemma: Stag Hunt has a fully cooperative equilibrium the PD lacks, and in Chicken, where the equilibria are off-diagonal, backing down is at least a best reply to aggression rather than a mistake. Two equilibria does not mean the players are indifferent: in Battle of the Sexes they actively disagree about which to pick. And no pure Nash does not mean unsolvable: Matching Pennies has a mixed equilibrium, which is where the next lesson goes.
Work it out
Classify Stag Hunt, then Matching Pennies
Stag Hunt. Mark best responses. If your partner hunts Stag, your best reply is Stag (4 > 3); if they take the Hare, your best reply is Hare (3 > 0). So both (Stag, Stag) and (Hare, Hare) are Nash. Both players prefer (Stag, Stag) at , yet Hare guarantees no matter what the partner does, while Stag risks . That safety is why groups can get stuck at the worse equilibrium.
Matching Pennies. Mark best responses again. Whatever cell you point at, one player always wants to switch: the matcher chases, the mismatcher flees. The best-response marks never coincide, so there is no pure Nash. The only stable way to play is to randomize, which is the mixed strategy of the next lesson.
Key idea
The same 2x2 shape holds a family of games. Read the payoffs for where the temptation sits and whether players want to match, and the equilibrium structure, one, two, or none, follows.
Check yourself
Sources
- Rapoport, A., and Guyer, M. (1966). A taxonomy of 2x2 games. General Systems, 11, 203-214.
- Osborne, M. J. (2004). An Introduction to Game Theory. Oxford University Press.