- Content map: SMU H3 Game Theory Map
Chapter 4: Simultaneous-Move Games with Discrete Strategies
Simultaneous-Move Games
Definition:
A simultaneous-move game is a game in which players choose strategies without observing the others’ realised choices.
- Strategic uncertainty is central: each player acts against expectations about what the others will do.
- In this part of the course, the strategy sets are finite, so the game is naturally written in matrix form.
- The main questions are which strategies survive rationality and which profiles are stable.
Insight:
Simultaneous play turns strategic prediction into a problem of mutual expectations.
Payoff Matrix (Strategic Form)
Definition:
The strategic form of a game lists each player’s strategies and the payoff attached to every strategy profile.
- One player’s strategies are written in rows and the other’s in columns.
- Each cell records the payoffs generated by that pair of actions.
- Strategic form lets us read off incentives directly.
- Dominance, best responses, and Nash equilibrium are all matrix-based ideas.
Dominant and Dominated Strategies
There are a few ways of defining whether a strategy is dominant or dominated.
Strict Dominance
Definition:
If every payoff in Strategy A is higher than every payoff in Strategy B for every action taken by the opponent:
- Strategy A strictly dominates Strategy B
- Strategy B is strictly dominated by Strategy A
Weak Dominance
Definition:
If every payoff in Strategy A is at least equal or higherx than every payoff in Strategy B for every action taken by the opponent:
- Strategy A weakly dominates Strategy B
- Strategy B is weakly dominated by Strategy A
- To test dominance, fix the opponent’s action and compare own payoffs across strategies.
- If a dominant strategy exists, a rational player uses it.
- Weak dominance means never worse and sometimes better.
Result:
If every player has a dominant strategy, the profile of dominant strategies is a Nash equilibrium.
Iterated Deletion of Dominated Strategies
- Rational players never choose dominated strategies.
- A game may contain dominated strategies even when no player has a dominant strategy.
- After one dominated strategy is deleted, new dominated strategies may appear in the reduced game.
- Iterated deletion of dominated strategies repeats this logic until no further dominated strategy remains.
Insight:
Iterated deletion formalises higher-order rationality: I rule out strategies that you should never play, then I re-evaluate what is left.
Best Responses
Definition:
A best response is a strategy that maximises a player’s payoff given the strategies chosen by the other players.
- Fix the opponent’s strategy.
- Compare the player’s payoffs across feasible strategies.
- Mark the payoff-maximising choices.
- Best-response reasoning becomes essential when dominance alone does not solve the game.
Nash Equilibrium
Definition:
A Nash equilibrium is a strategy profile where no player is incentivised to deviate from the equilibrium point.
Insight:
A Nash equilibrium is a strategy profile in which each player’s strategy is a best response to the strategies of the others.
- Nash equilibrium is the standard solution concept for finite simultaneous-move games.
- No player can gain from unilateral deviation.