Dominant Strategy

A dominant strategy is one that yields a weakly (or strictly) higher payoff than any alternative, no matter what the other players do. Strictly dominant: $u_i(s_i^*, s_{-i}) > u_i(s_i, s_{-i})$ for all $s_i \ne s_i^*$ and all $s_{-i}$. Weakly dominant: Replace $>$ with $\ge$, with strict inequality for at least one $s_{-i}$. Concrete example — Vickrey (second-price) auction: Each bidder submits a sealed bid. The highest bidder wins but pays the second-highest bid. Claim: bidding your true value $v$ is weakly dominant. If you bid $b > v$: you win more often, but when you win with the second-highest bid above $v$, you pay more than the item is worth — a loss. If you bid $b < v$: you sometimes lose auctions you would have profited from. Bidding $b = v$ avoids both pitfalls. Iterated elimination of dominated strategies (IEDS): If a strategy is dominated, a rational player never uses it. Remove it. Now in the reduced game, other strategies may become dominated. Remove those too. Repeat. If this process yields a unique outcome, the game is dominance solvable. Concrete example — Beauty Contest: Everyone picks a number from 0 to 100. The winner is closest to $\frac{2}{3}$ of the average. No rational player picks above 67 (since $\frac{2}{3}$ of 100 = 67). But if no one picks above 67, no one should pick above 44. Iterating: the unique IEDS outcome is 0. When to use: Always check for dominant strategies first — they are the simplest solution concept. Second-price auctions, prisoner's dilemma, and p-beauty contests are classic examples. If dominance does not resolve the game, move to Nash equilibrium analysis.