Unit 8: Game Theory

Game theory is the formal study of strategic interaction, where the outcome for each decision-maker depends not only on their own choices but on the choices of others. Developed by John von Neumann and Oskar Morgenstern (Theory of Games and Economic Behavior, 1944) and extended by John Nash (1950), it gives managerial economics a framework for pricing wars, entry decisions, bidding, and bargaining under interdependence.

• Player: any decision-making unit — a firm, a manager, a bidder — assumed to be rational and to know the payoff structure.
• Strategy: a complete plan of action a player will follow; a pure strategy picks one action with certainty, a mixed strategy randomises over actions.
• Payoff: the reward (profit, utility) a player receives for a combination of strategies, shown in a payoff matrix.
• Rationality and common knowledge: each player maximises their own payoff and knows that all others do the same, and this is known to all.
• Equilibrium: a stable outcome from which no player wishes to deviate unilaterally.

II. Meaning and Types of Games

The vocabulary of strategic situations and the categories economists sort them into.

A. Meaning of a game

A game is any situation of strategic interdependence formalised by players, strategies, and payoffs.

• Interdependence: one firm's best price depends on its rival's price, so it cannot be chosen in isolation.
• Normal (strategic) form: the game written as a payoff matrix, listing each player's strategies and the resulting payoffs, e.g. rows for Firm A, columns for Firm B.
• Extensive form: the game drawn as a decision tree showing the sequence of moves and information available at each node.

B. Types of games

Games are classified by timing, payoff structure, repetition, and information.

• Cooperative vs non-cooperative: in cooperative games players form binding, enforceable agreements (a cartel with a contract); in non-cooperative games each acts alone (secret price cuts), which is the standard managerial case.
• Simultaneous vs sequential:
  • Simultaneous: players move at the same time without observing others, best shown in normal form (two firms setting price for the season).
  • Sequential: one player moves after observing another, shown in extensive form and solved by backward induction (an incumbent reacting to entry).
• Zero-sum vs non-zero-sum: in zero-sum games one player's gain equals another's loss, so payoffs sum to zero (dividing a fixed market share); in non-zero-sum games total payoff varies, allowing mutual gain or mutual loss (advertising that grows the whole market).
• One-shot vs repeated: a one-shot game is played once; a repeated game is played many times, letting reputation and retaliation sustain cooperation.
• Perfect vs imperfect information: perfect information means every past move is observed (chess); imperfect means some moves are hidden (sealed bids).

III. Dominant Strategy and Nash Equilibrium

Two central solution concepts: the strategy that is always best, and the outcome no one wants to break.

A. Dominant strategy

A dominant strategy yields a higher payoff than any alternative regardless of what rivals do.

• Strict dominance: strategy S is strictly dominant if it gives a strictly greater payoff against every possible rival choice.
• Dominant strategy equilibrium: if every player has a dominant strategy, the combination of those strategies is the predicted outcome and needs no assumption about beliefs.
• Iterated elimination of dominated strategies: where no single dominant strategy exists, delete strategies that are never best; repeatedly applying this can narrow the game to one outcome.
• Worked example: two firms choose High or Low advertising.

                Firm B: High    Firm B: Low
Firm A: High      (10, 10)        (18, 6)
Firm A: Low       (6, 18)         (12, 12)

For Firm A, High beats Low against B-High (10 > 6) and against B-Low (18 > 12), so High is dominant. The matrix is symmetric, so High is dominant for B too, giving (High, High) → (10, 10).

B. Nash equilibrium

A Nash equilibrium is a set of strategies where each player's choice is a best response to the others', so no one gains by deviating alone.

• Best response: the strategy that maximises a player's payoff given fixed rival strategies; a Nash equilibrium is a mutual best response.
• Relation to dominance: every dominant strategy equilibrium is a Nash equilibrium, but many Nash equilibria exist without any dominant strategy.
• Existence and multiplicity: Nash (1950) proved every finite game has at least one equilibrium in pure or mixed strategies; some games have several pure-strategy equilibria (a coordination game where both firms adopt the same technical standard).
• Finding it in a matrix: underline each player's best-response payoff in every column and row; a cell where both payoffs are underlined is a Nash equilibrium.
• Managerial use: predicts the stable price, output, or capacity in an oligopoly — the Cournot and Bertrand outcomes are Nash equilibria.

IV. Prisoner's Dilemma

The classic demonstration that individually rational choices can produce a collectively poor result.

A. Structure and story

The prisoner's dilemma shows a dominant-strategy equilibrium that leaves both players worse off than an available alternative.

• The setup: two suspects are questioned separately; each may Confess (defect) or Stay Silent (cooperate). Payoffs are years in prison, so lower is better.

                    B: Silent      B: Confess
A: Silent           (-1, -1)        (-10, 0)
A: Confess          (0, -10)        (-5, -5)

• Dominant strategy: Confess beats Silent for A whether B stays silent (0 > -1) or confesses (-5 > -10), and symmetrically for B.
• Equilibrium vs optimum: the Nash equilibrium is (Confess, Confess) → (-5, -5), yet (Silent, Silent) → (-1, -1) is better for both. The gap is the dilemma.
• Why cooperation fails: each fears being the lone cooperator (the -10 "sucker" payoff) and is tempted by the defection gain (0), so mutual defection results.

B. Business applications

The dilemma models many managerial conflicts where mutual restraint would help but self-interest defeats it.

• Price wars: two firms would earn more by both holding high prices, but each has a dominant incentive to undercut, driving prices toward cost.
• Advertising and R&D races: rivals over-invest because cutting spending unilaterally cedes share, though joint restraint would raise profits.
• Cartel instability: OPEC-style output limits mirror (Silent, Silent); each member gains by secretly overproducing, so cartels tend to break down.

C. Escaping the dilemma

Cooperation can emerge when the game changes shape, chiefly through repetition.

• Repeated play: in an indefinitely repeated game, the threat of future retaliation can sustain cooperation, since a defection today is punished tomorrow.
• Tit-for-tat: cooperate first, then copy the rival's last move; Axelrod's tournaments found this simple, forgiving, retaliatory rule highly effective.
• Trigger strategies: a "grim" strategy cooperates until the first defection, then defects forever, making cheating unprofitable if firms value future profit enough.
• Enforcement and communication: binding contracts, most-favoured-customer clauses, or credible commitments convert the non-cooperative game into a cooperative one.

V. Mixed Strategy

Randomising over actions when no stable pure-strategy outcome exists.

A. Meaning and rationale

A mixed strategy assigns probabilities to a player's actions rather than choosing one for sure.

• Definition: player 1 plays action X with probability p and action Y with 1 − p; the opponent responds to these odds, not to a known move.
• When needed: games with no pure-strategy Nash equilibrium — typically pursuit or matching situations where each player wants to be unpredictable, such as an auditor deciding whom to inspect versus a firm deciding whether to comply.
• Mixed-strategy Nash equilibrium: a profile of probabilities where each player's mix is a best response to the other's mix; the existence theorem guarantees at least one such equilibrium in every finite game.

B. The indifference principle and solution

A player mixes so as to leave the rival indifferent between their own actions, otherwise the rival would switch to a pure strategy.

• Indifference condition: choose your probabilities so the opponent's expected payoff is equal across their actions, removing any reason for them to favour one.
• Worked example — Matching Pennies: each shows Heads or Tails; matching pays the matcher +1 and the mismatcher −1.

                B: Heads     B: Tails
A: Heads        (+1, -1)     (-1, +1)
A: Tails        (-1, +1)     (+1, -1)

There is no pure-strategy equilibrium. Let B play Heads with probability q. A's expected payoff from Heads is q(1) + (1 − q)(−1) = 2q − 1; from Tails it is 1 − 2q. Setting them equal gives q = ½, and by symmetry each player plays each action with probability ½.

• Expected payoff: at (½, ½) each player's expected payoff is 0, and neither can improve by changing the odds.

C. Managerial significance

Mixed strategies formalise deliberate unpredictability in competition and monitoring.

• Inspection and auditing: tax authorities and quality inspectors randomise audits so firms cannot predict and evade them.
• Competitive pricing and promotions: retailers randomise the timing and depth of discounts to stop rivals and consumers from anticipating sales.
• Interpretation of probabilities: the equilibrium mix can be read as the frequency of an action across a population of firms, not only as one manager's coin-flip, which makes the concept empirically usable.