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Strict dominance
A strategy strictly dominates another when it pays more against every strategy of the opponent. A player with a dominating strategy has no need to guess what the other will do.
Lesson · Games and the Social Contract · One-shot games
The Prisoner's Dilemma
In the prisoner's dilemma one strategy is strictly better than the other against every move of the opponent, so both players take it, and the cell they reach pays each less than the cell they passed over. The dilemma does not come from distrust; it is in the payoffs, and it goes away only when the payoffs change.
defect beats cooperate against either move
so both defect, and (D, D) is the cell
(C, C) pays both more than (D, D) does
the dilemma is in the payoffs, not the mood
Dominance, then the gap
Core ideas
What this lesson establishes
One definition, one argument that uses it, and one comparison that makes the argument uncomfortable.
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The equilibrium of a dominance argument
If both players have a strictly dominating strategy, the pair of them is the only cell where both are best-responding. Nothing else in the table survives.
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Pareto domination
A cell Pareto-dominates another when it pays someone more and nobody less. In the dilemma the cell nobody chooses pays both players more than the cell everybody chooses.
The material
Why the best move for each is the worst for both
The argument is four comparisons, and then a fifth that goes the other way.
Two people are held separately and each is asked to stay silent or to inform on the other. The standard payoffs are written C for cooperating with the other prisoner (staying silent) and D for defecting (informing). The four payoffs are named: R for the reward when both cooperate, T for the temptation to defect against a cooperator, P for the punishment when both defect, and S for the sucker's payoff of cooperating against a defector.
column C column D
row C 3, 3 0, 5
row D 5, 0 1, 1
Here R = 3, S = 0, T = 5 and P = 1, and the game is a prisoner's dilemma exactly when T > R > P > S. The ordering is the whole definition: temptation beats reward, reward beats punishment, punishment beats being the sucker.
Definition · Strict dominance
Strategy A strictly dominates strategy B for a player when A pays more than B against every strategy the other player might choose.
Take the row player. If the column player cooperates, D pays 5 and C pays 3. If the column player defects, D pays 1 and C pays 0. D is better in both cases, by 2 in the first and by 1 in the second, so D strictly dominates C. The table is symmetric, so the same holds for the column player.
That settles the play without asking what either player believes. A player who is certain the other will cooperate should defect, since 5 beats 3. A player who is certain the other will defect should defect, since 1 beats 0. Each, then, defects, and the cell reached is D, D.
Definition · Pareto domination
Cell X Pareto-dominates cell Y when every player gets at least as much in X as in Y, and at least one gets more.
Compare the cell reached with the one passed over. Mutual cooperation gives each player 3 and mutual defection gives each 1. So the only equilibrium of the game is Pareto-dominated, and the lab says so in the efficiency tile. That is the dilemma: each has a reason that does not depend on the other, and the reasons together defeat both.
What changes the game
The dilemma needs the ordering and nothing else. Lower the temptation from 5 to 4 and the ordering still holds, so D still dominates and the diagnosis stays the same. Set the reward at 4 and the temptation at 3, and against a cooperator cooperating now pays 4 where defecting pays 3. The ordering is broken, D no longer dominates, and mutual cooperation becomes an equilibrium alongside mutual defection.
The lab does not say which of these payoffs describes a real pair of prisoners, or whether real prisoners would not also care about each other. That would be a different table, and the computation would be on it.
Interactive
A game in a table
best responses, equilibria, dominance and efficiency, computed from the payoffs
Method
Showing a game is a dilemma
Four checks, in this order. A failure at any one means the game is something else.
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Compare the row player's two payoffs in each column
If the same strategy is larger in every column, it strictly dominates. If it is larger in one and tied or smaller in the other, it does not.
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Do the same for the column player
Compare along each row. A dilemma needs the dominating strategy for both players.
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Name the cell the two dominating strategies reach
That cell is the only one in which both players are best-responding; no belief about the other changes it.
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Compare it with the other cells
If some cell pays both players more, the equilibrium is Pareto-dominated. That cell, not the equilibrium, is what the players would choose together.
Dominance and the gap
Score 0 / 0Choose the answer you can justify.
The standard dilemma, line by line
Payoffs R = 3, S = 0, T = 5, P = 1.
Row vs column C: D pays 5, C pays 3 -> D better
Row vs column D: D pays 1, C pays 0 -> D better
Column, by symmetry: -> D better
Only equilibrium: (D, D), payoffs 1, 1
Passed over: (C, C), payoffs 3, 3
Cooperating together gives each player 3 against 1. The shortfall is 2 for each, and no one is able to close it alone: a player who moves to C while the other holds D drops to 0.
Common mistakes
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Thinking the dilemma comes from distrust, and would vanish between people who trust each otherDefect dominates, and a dominating strategy is best whatever the player believes about the other. Against a certain cooperator defecting pays 5 and cooperating 3, so trust makes defecting more tempting, not less. What can dissolve the dilemma is a different table, such as the one with reward 4 and temptation 3: there cooperating against a cooperator pays more than defecting, which is to say the players value the outcomes differently.
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Reading the equilibrium as the best the pair can doIt is the cell where each is best-responding, which is a different thing. The cell passed over gives both players 3 against the 1 they reach, and the dilemma is that gap.
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Treating “defect” as a verdict on the playersBoth strategies are names for columns in a table. The argument takes no view on who is wicked; it takes the payoffs as given and shows what follows.
Check your understanding
✓
Finish when you can prove a game is a dilemma, or show it is not.
Given a two-by-two table, check dominance for each player, name the cell the dominating strategies reach, and say whether another cell Pareto-dominates it. Then change one payoff so the verdict flips and say which.
Note: The cooperate-or-defect labels are conventional and the payoffs here are the usual teaching numbers; only the ordering of the four matters for the diagnosis. The prisoner's dilemma returns three times in this course: repeated, among many players, and as the shape of Hobbes's state of nature.