20 Rational Choice and Nash Equilibria
The Nash equilibrium (Chapter 19) is a fundamental tool in game-theoretic analysis across philosophy, political science, and economics. But there remains a foundational question: Why should rational players play their part in a Nash equilibrium? What makes this solution concept compelling for strategic interaction? The answer is less obvious than the concept’s ubiquity might suggest.
20.1 Nash Equilibrium as Self-Enforcing Agreements
One prominent justification for Nash equilibrium invokes the concept of self-enforcing agreements. The argument claims that Nash equilibria are the only strategy combinations players could agree upon without external enforcement, since no player would want to deviate once play begins.
A self-enforcing agreement is an agreement that provides sufficient incentive for all players to honor it without external enforcement mechanisms. This means players have reason to stick to the agreed-upon strategies based on the structure of the game and the nature of the agreement itself.
However, the relationship between Nash equilibria and self-enforcing agreements is more complex than this justification suggests. Consider the following game:
| Column | ||||
| \(l\) | \(c\) | \(r\) | ||
| Row | \(u\) | 4, 6 | 5, 4 | 0, 0 |
| \(m\) | 5, 7 | 4, 8 | 0, 0 | |
| \(d\) | 0, 0 | 0, 0 | 1, 1 | |
The strategy profile \((d, r)\) is a Nash equilibrium—neither player can improve by deviating. Yet this equilibrium yields payoffs of only \(1\) to each player, while the outcomes \((u, l)\), \((u, c)\), \((m, l)\), or \((m, c)\) would give both players payoffs of at least \(4\).
This means that an agreement to play the Nash equilibrium \((d, r)\) is not self-enforcing. Both players would rather abandon such an agreement and face the uncertainty of uncoordinated play—with the possibility of achieving much higher payoffs in the upper-left region—than commit to the certain but minimal payoff of \(1\) from \((d,r)\). The Nash equilibrium may be so unappealing that rational players would prefer to reject it entirely and take their chances without any agreement, rather than commit to it.
Conversely, consider this game:
| Column | |||
| \(l\) | \(r\) | ||
| Row | \(u\) | 0, 0 | 4, 2 |
| \(d\) | 2, 4 | 3, 3 | |
The profile \((d, r)\) with payoffs \((3, 3)\) is not a Nash equilibrium—Row prefers to deviate to \(u\) for a payoff of \(4\). However, \((d, r)\) might be self-enforcing if players agree to it. Both players get equal, reasonable payoffs. While each could get \(4\) by deviating unilaterally, the symmetry and mutual benefit of the \((d, r)\) outcome could provide sufficient reason for both players to honor their agreement, making it self-enforcing despite not being a Nash equilibrium.
These examples reveal that:
- Some Nash equilibria are not self-enforcing: They may be stable against unilateral deviation but so undesirable that rational players would refuse to coordinate on them.
- Some self-enforcing agreements are not Nash equilibria: Outcomes with sufficient mutual benefit and fairness might provide reason to honor an agreement despite the possibility of profitable deviation.
20.2 Rationalizability
While Nash equilibrium requires that all players’ beliefs about each other are correct, rationalizability captures a weaker notion: what strategies could rational players choose based on some reasonable belief about what others might do?
A strategy is rationalizable if there exists some belief about opponents’ strategies that makes it a best response. Crucially, those opponents’ strategies must themselves be best responses to some beliefs, and so on. This creates a chain of reasoning: “I play this because I think you’ll play that, which makes sense if you think I’ll play something else…”
To illustrate, consider the following game:
| Column | ||||
| \(l\) | \(c\) | \(r\) | ||
| Row | \(u\) | \(3, 2\) | \(0, 0\) | \(2, 3\) |
| \(m\) | \(0, 0\) | \(1, 1\) | \(0, 0\) | |
| \(d\) | \(2, 3\) | \(0, 0\) | \(3, 2\) | |
In this game, \((m, c)\) is the unique Nash equilibrium. Yet Row might rationally play \(d\) if she believes Column will play \(r\) (getting a payoff of \(3\) instead of \(0\) or \(2\)). Column might rationally play \(l\) if she believes Row will play \(d\) (getting a payoff \(3\) instead of \(0\) or \(2\)). This would result in the outcome \((d, l)\): Row’s belief about Column was wrong, but Column’s belief about Row was correct. Both players acted rationally given their beliefs, even though they didn’t reach the Nash equilibrium.
Not every strategy is rationalizable. A strategy that is strictly dominated—always worse than another strategy regardless of what opponents do—can never be a best response to any belief. Rational players will not play strictly dominated strategies, and knowing this, other players won’t expect them to.
Consider this game:
| Column | |||
| \(l\) | \(r\) | ||
| Row | \(u\) | \(5, 5\) | \(-100, 4\) |
| \(d\) | \(0, 1\) | \(0, 0\) | |
For Column, playing \(l\) strictly dominates playing \(r\): whether Row plays \(u\) or \(d\), Column always gets a higher payoff from \(l\) (\(5 > 4\) when Row plays \(u\), and \(1 > 0\) when Row plays \(d\)). No rational Column player would ever choose \(r\), and Row knows this. Given that Column will play \(l\), Row’s best response is \(u\), yielding a payoff of \(5\) rather than \(0\). Thus, Row can play \(u\) without worrying that they will get the really bad payoff of \(-100\).