Module 13 of 15

Game Theory

How rational choices by each player can produce outcomes no one wants.

AMathematical Foundation

A normal-form game lists players, their strategies and the payoff each player receives for every combination of strategies. A best response is a strategy that maximises a player's payoff given the others' strategies. A Nash equilibrium is a strategy profile in which every player is playing a best response, so no one gains by deviating alone. A strategy is strictly dominant if it is better than every alternative whatever the other player does. In a prisoner's dilemma, each player has a dominant strategy, yet the resulting equilibrium leaves both worse off than mutual cooperation.

(s_1^*,s_2^*)\text{ is a Nash equilibrium} \iff u_1(s_1^*,s_2^*)\ge u_1(s_1,s_2^*)\;\forall s_1\;\text{ and }\;u_2(s_1^*,s_2^*)\ge u_2(s_1^*,s_2)\;\forall s_2

In words: A pair of strategies is a Nash equilibrium if neither player can raise their own payoff by changing only their own strategy.

$u_i$: payoff function of player i
$s_i$: strategy of player i

q^*=\frac{d-b}{a-b-c+d}

In words: In a two by two game with row payoffs a, b, c, d, the column player's mixing probability that makes the row player indifferent is d minus b over a minus b minus c plus d.

$q^*$: probability that the column player plays column 1 which leaves the row player indifferent between rows
$a,b,c,d$: row player's payoffs in cells (1,1), (1,2), (2,1), (2,2)

Assumptions

Players know the payoffs and choose to maximise them.
The mixing formula applies only when its denominator is not zero and the resulting q lies strictly between 0 and 1; otherwise the game has no fully mixed equilibrium of this form.
The basic model is one-shot; repeated interaction, reputation and communication can change outcomes.

BIntuitive Explanation

Two rival football clubs could each invest patiently in their academies. If both do, both benefit. But if one buys ready-made players while the other waits, the buyer wins now. Whatever the rival does, buying looks better in the short run, so both buy, and both end up worse off than if they had both invested. A rule that binds both clubs changes the game.

CConnection to Debate

Game theory turns vague claims about incentives into testable structure. In the homegrown-players motion, the Proposition can argue that a quota solves a collective-action problem; the Opposition can argue that clubs will respond strategically (registering nominal homegrown players, hoarding youth talent). In luxury consolidation, conglomerates and independent suppliers bargain under unequal outside options. The key questions: who are the players, what are their payoffs, and how does the proposed rule change them?

DReal-World Application

Homegrown-player rules and club strategy
Competing interpretation
UEFA's squad-list rules reserve at least eight of 25 List A places for locally trained players; a club with fewer must register a smaller squad. Because the rule governs registration, not selection, clubs can comply strategically, for example by registering locally trained players who rarely play. A study for the European Commission found only very modest effects on competitive balance and very little effect on youth training. A starting-eleven quota, as proposed in the motion, would change clubs' payoffs more directly.
Sources
- UEFA Champions League league phase player lists: registrationUEFA, 2026-09-03
- Study on the Assessment of UEFA's Home Grown Player Rule (final report)University of Liverpool and Edge Hill University, for the European Commission, 2013-04-30
Partnerships and pressure in advocacy
Competing interpretation
The EDF–McDonald's partnership began after public pressure on the company in the late 1980s. A bargaining model treats that pressure as part of the environmental group's leverage: a company cooperates when partnering costs less than facing continued campaigns. If a group commits to partnership and gives up campaigning, its outside option weakens and the company's incentive to concede falls. Whether this happens in practice is an empirical question the model cannot settle.
Sources
- McDonald's saves billions cutting waste (EDF partnership case study)Environmental Defense Fund

These examples describe real policies and research. The numbers in the interactive below and in the practice questions are hypothetical teaching data unless a question says otherwise.

EInteractive Visualization

Edit a 2×2 payoff matrix and see best responses, dominant strategies, pure Nash equilibria and the mixed equilibrium.

Illustrative example — hypothetical data.Stylised payoffs (higher is better). Edit any number to build your own game.

Game

Two rival clubs. Developing youth is better for both if both do it, but each gains a short-term edge by buying instead.

Payoff matrix: first number is the row player’s payoff, second the column player’s
	Column: Invest in academy	Column: Buy ready-made players
Row: Invest in academy	,	,
Row: Buy ready-made players	,	, Nash equilibrium

Bold underlined numbers are best responses: the best payoff a player can get given the other’s choice. A cell where both numbers are best responses is a Nash equilibrium.

Pure equilibria

Dominant strategy (row)

Buy ready-made players

Dominant strategy (column)

Buy ready-made players

Mixed equilibrium

None

Stable outcome: Buy ready-made players / Buy ready-made players. Every equilibrium leaves both players worse off than another outcome they could reach together. Individual incentives produce a collectively worse result, the classic argument for a rule that binds everyone.

Equilibrium analysis assumes players know the payoffs and act on them. Real clubs, firms and regulators also care about reputation, repeated play and rules that change the payoffs.

FDebate Practice

GLimitations

Real payoffs are rarely known, players are not perfectly rational, and games are usually repeated with many players. Models are useful for revealing incentive structures, but the conclusion is only as good as the payoffs assumed, which should be defended with evidence.

AMathematical Foundation

(s_1^*,s_2^*)\text{ is a Nash equilibrium} \iff u_1(s_1^*,s_2^*)\ge u_1(s_1,s_2^*)\;\forall s_1\;\text{ and }\;u_2(s_1^*,s_2^*)\ge u_2(s_1^*,s_2)\;\forall s_2

In words: A pair of strategies is a Nash equilibrium if neither player can raise their own payoff by changing only their own strategy.

$u_i$: payoff function of player i
$s_i$: strategy of player i

q^*=\frac{d-b}{a-b-c+d}

In words: In a two by two game with row payoffs a, b, c, d, the column player's mixing probability that makes the row player indifferent is d minus b over a minus b minus c plus d.

$q^*$: probability that the column player plays column 1 which leaves the row player indifferent between rows
$a,b,c,d$: row player's payoffs in cells (1,1), (1,2), (2,1), (2,2)

Assumptions

Players know the payoffs and choose to maximise them.
The mixing formula applies only when its denominator is not zero and the resulting q lies strictly between 0 and 1; otherwise the game has no fully mixed equilibrium of this form.
The basic model is one-shot; repeated interaction, reputation and communication can change outcomes.

BIntuitive Explanation

CConnection to Debate

DReal-World Application

Homegrown-player rules and club strategy
Competing interpretation
UEFA's squad-list rules reserve at least eight of 25 List A places for locally trained players; a club with fewer must register a smaller squad. Because the rule governs registration, not selection, clubs can comply strategically, for example by registering locally trained players who rarely play. A study for the European Commission found only very modest effects on competitive balance and very little effect on youth training. A starting-eleven quota, as proposed in the motion, would change clubs' payoffs more directly.
Sources
- UEFA Champions League league phase player lists: registrationUEFA, 2026-09-03
- Study on the Assessment of UEFA's Home Grown Player Rule (final report)University of Liverpool and Edge Hill University, for the European Commission, 2013-04-30
Partnerships and pressure in advocacy
Competing interpretation
The EDF–McDonald's partnership began after public pressure on the company in the late 1980s. A bargaining model treats that pressure as part of the environmental group's leverage: a company cooperates when partnering costs less than facing continued campaigns. If a group commits to partnership and gives up campaigning, its outside option weakens and the company's incentive to concede falls. Whether this happens in practice is an empirical question the model cannot settle.
Sources
- McDonald's saves billions cutting waste (EDF partnership case study)Environmental Defense Fund

These examples describe real policies and research. The numbers in the interactive below and in the practice questions are hypothetical teaching data unless a question says otherwise.

EInteractive Visualization

Edit a 2×2 payoff matrix and see best responses, dominant strategies, pure Nash equilibria and the mixed equilibrium.

Illustrative example — hypothetical data.Stylised payoffs (higher is better). Edit any number to build your own game.

Game

Two rival clubs. Developing youth is better for both if both do it, but each gains a short-term edge by buying instead.

Payoff matrix: first number is the row player’s payoff, second the column player’s
	Column: Invest in academy	Column: Buy ready-made players
Row: Invest in academy	,	,
Row: Buy ready-made players	,	, Nash equilibrium

Bold underlined numbers are best responses: the best payoff a player can get given the other’s choice. A cell where both numbers are best responses is a Nash equilibrium.

Pure equilibria

Dominant strategy (row)

Buy ready-made players

Dominant strategy (column)

Buy ready-made players

Mixed equilibrium

None

Equilibrium analysis assumes players know the payoffs and act on them. Real clubs, firms and regulators also care about reputation, repeated play and rules that change the payoffs.

FDebate Practice