ECON5001 · Microeconomic Theory
Game Theory
Game theory is where ECON5001 turns strategic: your best move now depends on what a rival does, and they are reasoning about you at the same time. The toolkit is compact — describe the game (players, strategies, payoffs), then apply solution concepts in order of strength: dominance and iterated elimination, then Nash equilibrium (every player best-responding, no profitable unilateral deviation), and for sequential games subgame-perfect equilibrium found by backward induction. The signature exam skills are the best-response (underline) sweep that finds every pure-strategy Nash equilibrium in a payoff matrix, and backward induction on a game tree, which discards non-credible threats. Master these and the oligopoly chapter (Cournot, Stackelberg, Bertrand) becomes the same logic with continuous strategies.
What this chapter covers
- 011. Describing a game — players i, strategy sets Sᵢ, payoffs Uᵢ(sᵢ, s₋ᵢ); rational and intelligent players
- 022. Normal (strategic) form — the payoff matrix for static (simultaneous) games
- 033. Extensive form — the game tree for dynamic (sequential) games
- 044. Dominance — strict vs weak; a dominant strategy is best whatever others do
- 055. IEDS — iterated elimination of dominated strategies; dominance-solvable games
- 066. Best response and Nash equilibrium — mutual best responses, no profitable deviation
- 077. The best-response sweep — underline best replies to find every pure-strategy NE
- 088. Dynamic games — subgames, subgame-perfect equilibrium and backward induction
Find the dominant strategies and all Nash equilibria of a coordination game
- +1(a) Check Alex (compare Alex's own payoff — the first number — within each of Sam's columns). If Sam plays Cafe: Cafe gives 3 > 0 from Bar → Alex prefers Cafe. If Sam plays Bar: Bar gives 2 > 1 from Cafe → Alex prefers Bar.
- +1(a) Alex's best move flips with Sam's choice, so Alex has NO dominant strategy. By the same comparison on the second number, Sam prefers Cafe when Alex plays Cafe (2 > 0) and Bar when Alex plays Bar (3 > 1) → Sam has no dominant strategy either.
- +1(b) Alex's best responses: to Sam = Cafe → Cafe; to Sam = Bar → Bar. (Underline Alex's best first-number in each column.)
- +1(b) Sam's best responses: to Alex = Cafe → Cafe (2 > 1); to Alex = Bar → Bar (3 > 0). (Underline Sam's best second-number in each row.)
- +1(b) A cell is a Nash equilibrium only where BOTH are best-responding. (Cafe, Cafe): Alex best to Cafe ✓, Sam best to Cafe ✓ → NE = (3, 2). (Bar, Bar): Alex best to Bar ✓, Sam best to Bar ✓ → NE = (2, 3).
- +1(b) Check the off-diagonal cells: (Cafe, Bar) and (Bar, Cafe) each fail on at least one side, so they are not equilibria. Two pure-strategy Nash equilibria in total.
Key terms
- Normal (strategic) form
- A payoff matrix representing a static (simultaneous) game: rows are one player's strategies, columns the other's, and each cell holds the payoff pair (row player, column player). The extensive form (game tree) is its counterpart for sequential games.
- Dominant strategy
- A strategy that gives a player a higher payoff than every alternative no matter what opponents do. Strict dominance requires strictly greater payoffs everywhere; weak dominance allows ties (≥ always, > somewhere). If every player has a dominant strategy, that profile is automatically the Nash equilibrium.
- IEDS (iterated elimination of dominated strategies)
- Repeatedly delete dominated strategies; in the smaller game new strategies may become dominated, so iterate. If one strategy each remains the game is 'dominance solvable'. Any survivor profile is a Nash equilibrium, but IEDS need not find every Nash equilibrium — and only strict dominance is safe to iterate.
- Best response
- The strategy (or strategies) that maximise a player's payoff given a fixed choice by the others, written Rᵢ(s₋ᵢ). A Nash equilibrium is exactly a profile in which everyone is simultaneously playing a best response.
- Nash equilibrium
- A strategy profile s* in which no player can gain by a unilateral deviation: Uᵢ(sᵢ*, s₋ᵢ*) ≥ Uᵢ(sᵢ, s₋ᵢ*) for all sᵢ and all i. Found by the best-response (underline) sweep, by IEDS, or by intersecting reaction functions in continuous games.
- Prisoner's Dilemma
- A game in which each player has a dominant strategy to defect, so the unique Nash equilibrium is mutual defection — yet it is Pareto-dominated by mutual cooperation. The classic illustration that individual rationality need not give a collectively efficient outcome, and why cartels are unstable.
- Mixed strategy
- A probability distribution over a player's pure strategies. Nash's theorem guarantees every finite game has at least one equilibrium, possibly in mixed strategies — so when the best-response sweep finds no pure-strategy cell (as in Rock-Paper-Scissors), the equilibrium is mixed.
- Subgame-perfect equilibrium (SPE)
- A refinement for dynamic games: a strategy profile that is a Nash equilibrium in every subgame, found by backward induction (solve the last mover first, fold payoffs back). It rules out non-credible threats that ordinary Nash equilibrium would allow.
Game Theory FAQ
What is the difference between a dominant strategy and a Nash equilibrium?
A dominant strategy is best for one player regardless of what anyone else does — a property of a single strategy. A Nash equilibrium is a whole profile in which every player is best-responding to the others, so no one can gain by deviating alone. Dominant strategies are stronger but rare; if each player happens to have one, the dominant-strategy profile is automatically a Nash equilibrium, but most games have Nash equilibria without any dominant strategy.
How do I find every pure-strategy Nash equilibrium in a payoff matrix?
Use the best-response (underline) sweep. In each column, underline the row player's largest first-number; in each row, underline the column player's largest second-number. Any cell where both numbers are underlined is a Nash equilibrium. This reliably catches zero, one, or several equilibria with no guesswork. If no cell qualifies, the equilibrium is in mixed strategies.
Why is the Prisoner's Dilemma outcome called inefficient?
Because both players would be better off at the cooperative outcome, yet each has a dominant strategy to defect, so the unique Nash equilibrium is mutual defection — a worse payoff for both. The cooperative outcome is Pareto-superior but is not an equilibrium, since from there either player gains by defecting. The exam mark is in naming the equilibrium AND stating it is Pareto-inefficient; do not call the cooperative cell the equilibrium.
What is a non-credible threat, and why do we need backward induction?
In a sequential game, a plain Nash equilibrium can rely on a player threatening an action they would never actually take once that point is reached (for example, an incumbent threatening to 'fight' an entrant when accommodating actually pays more). Backward induction — solving the last mover's choice first and folding payoffs back — produces the subgame-perfect equilibrium, which keeps only credible actions and discards such threats.
Does every game have a Nash equilibrium?
Yes. Nash's theorem guarantees that every finite game (finitely many players and strategies) has at least one Nash equilibrium, though it may be in mixed strategies rather than pure ones. So if your best-response sweep finds no pure-strategy equilibrium, you have not made an error — the equilibrium involves randomising, as in Rock-Paper-Scissors.
Does IEDS find all the Nash equilibria?
No. Every profile that survives iterated elimination of strictly dominated strategies is a Nash equilibrium, but the converse fails: a game can have Nash equilibria that IEDS never isolates. Use IEDS to shrink the game, then run the best-response sweep on what remains to capture every equilibrium. Also iterate only on strict dominance — eliminating on weak dominance can delete a genuine equilibrium.
Exam move
Treat game theory as a fixed routine rather than a set of facts, because the exam rewards solving small games fast. First classify the timing: simultaneous play means a payoff matrix and the solution concepts dominance and Nash equilibrium; sequential play means a game tree and backward induction. Then work the ladder in order of strength — look for dominant strategies and use iterated elimination (strict dominance only) to shrink the game, then run the best-response underline sweep to find every pure-strategy Nash equilibrium, always reporting all of them rather than the first you see. Drill the Prisoner's Dilemma until you can state both that mutual defection is the unique equilibrium and that it is Pareto-inefficient, and drill backward induction until you instinctively reject non-credible threats. Practise on the unit's problem sets and past papers with the clock running, redoing each example with fresh numbers after you read the method, and keep a short list of the traps that cost marks — assuming every equilibrium survives IEDS, iterating on weak dominance, calling the cooperative outcome the equilibrium, and accepting a non-credible threat.
