ECON90034 Chap.5 Simultaneous Games and Nash Equilibrium
Simultaneous Games and Nash Equilibrium
Game theory begins with players who choose at the same time. The subject teaches two solution ideas and asks you to keep them apart. An equilibrium in dominant strategies exists only when each player has a strategy that is strictly better whatever the others do; a Nash equilibrium is any list of strategies in which no player gains by switching alone, found by marking best responses.
The prisoners' dilemma shows individually rational choices producing an outcome both players dislike, while coordination games such as the battle of the sexes have two equilibria and no dominant strategy. Part A regularly asks you to find the Nash equilibria of a given game, and the review lecture notes that eliminating dominated strategies is not on the exam.
The chapter marks best responses on a three by three game, works an advertising dilemma between two banks, and shows how a change in costs can make the dilemma disappear.
What this chapter covers
- 01
Players, strategies and payoffs in a matrix
- 02
Strictly dominant and weakly dominant strategies
- 03
Equilibrium in dominant strategies
- 04
Best responses and Nash equilibrium
- 05
Games with several equilibria or none in pure strategies
- 06
The prisoners' dilemma and why cooperation is not enforceable
- 07
Coordination games: the battle of the sexes
- 08
Writing an equilibrium as a strategy profile
Worked example · free
A coordination game with two equilibria
- 1Row firm: against A, A pays 3 and B pays 1; against B, A pays 0 and B pays 2. Its best reply matches the other firm, so it has no dominant strategy.
- 1Column firm: against A, A pays 2 and B pays 0; against B, A pays 1 and B pays 3. It also matches, with no dominant strategy.
- 1Cells carrying both best replies: (A, A) with payoffs 3 and 2, and (B, B) with payoffs 2 and 3.
- 1Check (B, A): the row firm would switch to A and earn 3 instead of 1, so it is not an equilibrium.
Key terms
- Dominant strategy
- A strategy that gives a player a strictly higher payoff than any other, whatever the other players choose.
- Best response
- The strategy that gives a player the highest payoff against a given choice by the other players.
- Prisoners' dilemma
- A game in which each player has a dominant strategy, yet the resulting equilibrium leaves both worse off than an outcome they cannot enforce.
- Weakly dominant strategy
- A strategy that is at least as good as any other against every rival choice and strictly better against at least one; unlike a dominant strategy it can tie.
- Coordination game
- A game in which players gain from choosing matching actions, so it has several equilibria and the difficulty is agreeing on which one to play.
Simultaneous Games and Nash Equilibrium FAQ
What is the difference between a dominant strategy and a weakly dominant one?
A dominant strategy is strictly better against every move of the rival. A weakly dominant strategy is better or equal everywhere and strictly better somewhere, and a profile of weakly dominant strategies does not count as an equilibrium in dominant strategies.
How do you find all Nash equilibria quickly?
Mark the row player's best payoff in each column and the column player's best payoff in each row, including ties. Every cell with both marks is a Nash equilibrium, written as the strategies chosen.
Why do players in a prisoners' dilemma not cooperate?
Each player gains by defecting whatever the other does, and no agreement to cooperate can be enforced in a single play, so both defect and end up with less than mutual cooperation would give.
Can a game have no Nash equilibrium?
Yes, in pure strategies. If every cell of the matrix lacks a best-reply mark for at least one player, no pure strategy profile is stable. Mixed strategies, which would restore an equilibrium, are set aside in this subject, so the answer is simply that none exists in pure strategies.
Is eliminating dominated strategies on the exam?
The review lecture lists it as not examined. Best responses find every pure-strategy Nash equilibrium directly and are the method the exam expects, so practise that routine instead.
Exam move
Drill best-response marking on every matrix you meet until a three by three game takes under a minute. For each game, write the dominance check, the list of equilibria as strategy profiles, and one sentence interpreting the result, such as a dilemma or a coordination problem. Keep the subject's definitions word-perfect, since quiz questions turn on the difference between strict and weak dominance.
Keep a short list of classic shapes, the prisoners' dilemma, the coordination game and games with no pure equilibrium, and name the shape in one sentence whenever you finish a matrix.
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