Dominant Strategies and Nash Equilibrium

DOMINANT STRATEGIES & NASH EQUILIBRIUM how to actually SOLVE a game — find the stable outcome nobody wants to walk away from 1 · BEST RESPONSE & DOMINANCE BEST RESPONSE — your best move GIVEN what the others do. (varies with their choice.) DOMINANT strategy — best NO MATTER what they do. it beats your other options in every column. DOMINATED strategy — always worse than some other option → a rational player NEVER plays it. if you HAVE a dominant strategy, just play it — you don't even need to guess what they'll do. when BOTH players have one, their intersection is the outcome: a DOMINANT-STRATEGY EQUILIBRIUM. but many games have NO dominant strategy — for those we need a more general solution: the NASH EQUILIBRIUM. → 2 · FINDING NASH — the best-response method THEM Left Right YOU Up Down (4, 4) (3, 2) (2, 3) (1, 1) NASH EQUIL. underline each player's BEST payoff in every row/column. a cell where BOTH are underlined = a Nash equilibrium — here (Up, Left) = (4,4). each is already doing their best GIVEN the other, so neither wants to move. (this one's also a dominant-strategy equilibrium.) 3 · WHAT A NASH EQUILIBRIUM IS a set of strategies (one per player) where NO one can do better by changing ONLY their own move. = everyone is playing a BEST RESPONSE to everyone else. mutual best responses. a no-regret standstill. NE: (Up, Left) you 4 / them 4 you deviate → Down you'd get 2 < 4 → worse they deviate → Right they'd get 2 < 4 → worse NASH'S THEOREM (1950): EVERY finite game has at least one equilibrium — possibly in MIXED (randomised) strategies (board 05). so the solution always exists. John Nash won the 1994 Nobel for this. it is THE central solution concept of game theory. 4 · ITERATED ELIMINATION of dominated strategies (same game) THEM Left Right YOU Up Down (4, 4) (3, 2) (2, 3) (1, 1) 1) 'Down' is dominated for YOU (2<4, 1<3) → drop it. 2) then 'Right' is dominated for THEM (2<4) → drop it. 3) one cell left: (Up, Left) = (4,4). same answer as the underline method. ✓ keep deleting any strategy that's always worse than another. if you're left with ONE cell, that's the unique NE. (not every game collapses this far — then you fall back to underlining best responses to find the equilibria.) 5 · CAREFUL: NASH is STABLE, not necessarily GOOD THEM Coop Defect YOU Coop Defect (3, 3) (0, 5) (5, 0) (1, 1) NE better for BOTH! Defect dominates for each (5>3, 1>0) → the unique NE is (Defect, Defect) = (1,1). but (Coop, Coop) = (3,3) is better for EVERYONE. the stable outcome is the WORSE one. a NE is 'no one wants to deviate' — NOT 'best for the group'. this gap is the Prisoner's Dilemma → board 03. EQUILIBRIA IN SYSTEMS (shop / Agent) • 'DEFECT is dominant' is why unregulated shared resources degrade: everyone retries harder, floods the queue, and the stable state is the congested one — you must change the PAYOFFS (quotas, backoff) to move the equilibrium. • a stable BAD architecture is a Nash equilibrium: no single team can switch unilaterally without being worse off, so it persists even though everyone prefers the alternative. coordination (or a mandate) breaks it. • when you have a DOMINANT strategy (a choice that's best regardless of what competitors/users do), take it — you don't need to predict them. most 'what will they do?' agonising is over dominated options. JARGON, DECODED best response — your optimal move given the others' dominant strategy — best no matter what they do dominated strategy — always worse; never rational Nash equilibrium — mutual best responses; no one deviates dominant-strategy equilibrium — a NE from dominance iterated elimination — delete dominated moves repeatedly Pareto-optimal — can't help one without hurting another Nash's theorem — every finite game has ≥1 equilibrium unilateral deviation — one player changing only their move