From my doctoral course
This note comes from the doctoral course “Game Theory and Bargaining”. I rewrote the explanations in my own words and recomputed the numerical examples.
The prisoner’s dilemma
Two suspects are interrogated separately and each can stay silent or confess. (Payoff = years in prison with a minus sign; the figures are illustrative.)
| Silent | Confess | |
|---|---|---|
| Silent | (−1, −1) | (−10, 0) |
| Confess | (0, −10) | (−5, −5) |
If the other stays silent, confessing is better (0 instead of −1); and if the other confesses, confessing is better too (−5 instead of −10). Confessing is a dominant strategy; so both confess and get (−5, −5), whereas had both stayed silent they would have got (−1, −1). Individual rationality leads to a bad collective outcome; this is the paradox. I think it is the key pattern behind many of the “why do we wreck everything” moments in organisations.
Nash equilibrium
A combination of strategies is a Nash equilibrium if no player can improve their own position by unilaterally changing their strategy. An equilibrium is not necessarily the best outcome (in the prisoner’s dilemma the equilibrium outcome is not efficient), but it is “stable”.
How to find it: in the matrix, mark the row’s best response for each column and the column’s best response for each row; a cell with both marks is an equilibrium.
Dominant strategies and iterated elimination
- Strictly dominant: a strategy that is better than all the other options against every choice of the opponent. Weakly dominant: at least equal against all and better against at least one.
- Iterated elimination of strictly dominated strategies: if a strategy is always dominated, a rational player never chooses it; so remove it and check again with the smaller matrix.
Example (payoff in each cell: row player, column player):
| Col 1 | Col 2 | Col 3 | |
|---|---|---|---|
| Row 1 | (1, 1) | (2, 4) | (4, 5) |
| Row 2 | (5, 5) | (5, 1) | (5, 4) |
| Row 3 | (8, 8) | (8, 6) | (0, 6) |
- Row 1 is dominated by Row 2 (5>1, 5>2, 5>4) → removed.
- Now Column 2 is dominated by Column 1 (in rows 2 and 3: 5>1 and 8>6) → removed.
- Column 3 is also dominated by Column 1 (5>4 and 8>6) → removed.
- With Column 1 alone, Row 2 is dominated by Row 3 (5<8) → removed.
What remains is (Row 3, Column 1) with payoff (8, 8), the unique Nash equilibrium. Interesting: there was no dominant strategy at the start, but iterated elimination solved everything.
When the equilibrium is not pure or is multiple
- Multiple equilibria (like a coordination game): the “battle of the sexes” with payoffs (2,1) and (1,2) for the two possible coordinations and 0 for miscoordination has two pure equilibria and one mixed equilibrium: player 1 chooses their own preferred option with probability . The theory cannot predict which will happen; a social convention or a focal point is needed.
- No pure equilibrium: like “rock-paper-scissors”; the equilibrium exists only in mixed strategies. (Zero-sum games: maximin and mixed strategies)
Related: Game theory: introduction and history.