ECC1000 Chap.6 Strategic Interaction, Nash Equilibrium and Social Dilemmas
Strategic Interaction, Nash Equilibrium and Social Dilemmas
A different kind of constraint
Every model up to this point treated the environment as given while one decision-maker optimised. Week 4 drops that. Here the outcome depends on what someone else chooses and their outcome depends on you, so the constraint is another person's reasoning.
That needs its own apparatus: a way of describing the interaction, a way of predicting what each side will do, and a separate test of whether the predicted result is any good.
Five pieces of information make a game
A game specifies the players, the feasible strategies open to each, the order of play, the information each holds when choosing, and the payoffs attached to every combination of actions.
Leaving any one out breaks the analysis: a missing action can be the one that beats the predicted outcome, and without the full payoff grid no best response can be identified. The unit works with simultaneous games, in which players choose without observing each other, under an assumption of full information, meaning each knows the options and payoffs of both but not the choice being made.
Sequential play, where one moves after observing the other, is the workshop's territory, and order of play is the single element that separates the two treatments.
Solving a game is a two-pass scan
A best response is the action giving a player the highest payoff against one specific choice by the other.
Hold each of the column player's actions fixed and mark the row player's better payoff, then hold each of the row player's actions fixed and mark the column player's better payoff. Cells carrying both marks are Nash equilibria, written with the row player's action first.
The definition behind the procedure is worth restating: an equilibrium is a pair of actions from which neither player can improve their own payoff by changing only their own action.
Nothing in that requires the players to communicate, to agree, or to be satisfied with the result.
Two further tests, and why they are not the same test
Dominance asks whether one player would take the same action whatever the other did; a dominated action is one that is never taken. Pareto efficiency asks whether the resulting outcome could be improved for someone without harming anyone.
The two are independent: a player can hold a dominant strategy in a game whose equilibrium is inefficient, and a game can be efficient at its equilibrium with no dominant strategy anywhere in it.
Run dominance first where it exists, because it identifies the equilibrium without further work, then ask efficiency of the cell dominance delivered, testing every other cell against it rather than reasoning from the story.
The three shapes the unit puts on the grid
In a prisoner's dilemma both parties have an incentive to undercut, and the unique equilibrium is Pareto-dominated by the outcome cooperation would have produced; neither side can reach that outcome alone.
In a coordination game both do better by matching, so there are two equilibria, and the model narrows the possibilities without selecting between them. Sometimes one match is better for both, and the barrier is risk rather than preference; sometimes the two players prefer different matches and neither equilibrium dominates.
A public-goods game is the dilemma with more players: contributing carries a private cost while the benefit is shared regardless of who paid, so withholding is dominant and the group settles below what cooperation would have delivered. Because every player faces identical strategies and payoffs, the game is symmetric and can be drawn as one player against everyone else without losing the logic.
What this chapter covers
- 01
Why a new model is needed once outcomes interact
- 02
The five elements of a game
- 03
Simultaneous play and the full-information assumption
- 04
Best responses and the two-pass scan
- 05
Nash equilibrium and its notation
- 06
Dominant and dominated strategies
- 07
Pareto efficiency as a separate test
- 08
The prisoner's dilemma and coordination games
- 09
Free riding and the symmetric public-goods game
Solve a two-firm game and test the result
- 1Mark the row player's best response in each column.
- 1Mark the column player's best response in each row.
- 1Read the overlap and name any dominant strategy.
- 1Test the equilibrium cell for Pareto efficiency.
Key terms
- Strategic interaction
- A situation in which each participant's outcome depends on the actions of the others.
- Strategy
- An action, or set of actions, a player may choose in a game.
- Best response
- The action giving a player the highest payoff against one specific choice by the other.
- Nash equilibrium
- A combination of actions from which no player can gain by changing only their own action.
- Dominant strategy
- An action a player prefers regardless of what the other player does.
- Dominated strategy
- An action a player never prefers, whatever the other player does.
- Pareto efficiency
- A property of an outcome that cannot be improved for one player without worsening another.
- Public good
- Something whose benefit reaches everyone regardless of who paid for it.
- Symmetric game
- A game in which every player faces identical strategies and identical payoffs.
Strategic Interaction, Nash Equilibrium and Social Dilemmas FAQ
Does an equilibrium mean the outcome is good for the players?
No. The definition only says that no player can improve their own payoff by changing their own action alone. The freight and cafe games both settle on an outcome that both parties would gladly trade away, and that gap between stability and desirability is what social dilemmas are made of.
What should you conclude when a game has two equilibria?
That the analysis has narrowed the possibilities to two and stopped there. Both matched outcomes satisfy the definition, so predicting one of them requires something the model does not contain, such as a convention, communication or a commitment. Naming what would be needed is worth more than picking a winner.
Is a dominant strategy required for a game to have an equilibrium?
No. Dominance is a shortcut that identifies the equilibrium immediately when it exists, but plenty of games have none and are still solved by the two-pass scan. Absence of dominance narrows nothing on its own, which is why the scan rather than the dominance test is the primary method.
Why can a symmetric game be drawn with only two players?
Because every player faces the same strategies and the same payoffs, so the situation facing any one of them is fully described by their own choice and the behaviour of the rest taken together. Collapsing the group into a single opposing column preserves each payoff while keeping the grid readable.
What removes a free-riding problem?
Changing the payoffs so that withholding is no longer dominant. That can mean adding a private cost to withholding, adding a private reward for contributing, or restructuring the assessment so part of the outcome depends on individual effort. Nothing about the equilibrium changes until the numbers in the grid change.
Exam move
Solve every game with the scan, even when the answer looks obvious, and write the equilibrium in the grid's own order. Then run the two extra tests as separate sentences: dominance for each player, and efficiency for the equilibrium cell against every other cell. Finish by asking what would have to change in the payoffs to move the equilibrium, which is the question a policy version of the problem is really asking.
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