Simultaneous games

In game theory, a simultaneous game or static game is a game where each player chooses their action without knowledge of the actions chosen by other players.

Simultaneous games contrast with sequential games, which are played by the players taking turns (moves alternate between players). In other words, both players normally act at the same time in a simultaneous game. Even if the players do not act at the same time, both players are uninformed of each other’s move while making their decisions. Normal form representations are usually used for simultaneous games.

Given a continuous game, players will have different information sets if the game is simultaneous than if it is sequential because they have less information to act on at each step in the game.

For example, in a two player continuous game that is sequential, the second player can act in response to the action taken by the first player. However, this is not possible in a simultaneous game where both players act at the same time.

Characteristics

In sequential games, players observe what rivals have done in the past and there is a specific order of play. However, in simultaneous games, all players select strategies without observing the choices of their rivals and players choose at the exact same time.

A simple example is rock-paper-scissors in which all players make their choice at the exact same time. However moving at exactly the same time isn’t always taken literally, instead players may move without being able to see the choices of other players. A simple example is an election in which not all voters will vote literally at the same time but each voter will vote not knowing what anyone else has chosen.

Strategies - the best choice

Pure vs Mixed Strategy

Pure strategies are those in which players pick only one strategy from their best response. Mixed strategies are those in which players randomize strategies in their best responses set.

For simultaneous games, players will typically select mixed strategies while very occasionally choosing pure strategies. The reason for this is that in a game where players don’t know what the other one will choose it is best to pick the option that is likely to give the you the greatest benefit for the lowest risk given the other player could choose anything i.e. if you pick your best option but the other player also picks their best option, someone will suffer.

Dominant vs Dominated Strategy

A dominant strategy provides a player with the highest possible payoff for any strategy of the other players. In simultaneous games, the best move a player can make is to follow their dominant strategy, if one exists.

When analyzing a simultaneous game:

Firstly, identify any dominant strategies for all players. If each player has a dominant strategy, then players will play that strategy however if there is more than one dominant strategy then any of them are possible.

Secondly, if there aren’t any dominant strategies, identify all strategies dominated by other strategies. Then eliminate the dominated strategies and the remaining are strategies players will play.

Maximin Strategy

Some people always expect the worst and believe that others want to bring them down when in fact others want to maximise their payoffs. Still, nonetheless, player A will concentrate on their smallest possible payoff, believing this is what player A will get, they will choose the option with the highest value.

This option is the maximin move (strategy), as it maximises the minimum possible payoff. Thus, the player can be assured a payoff of at least the maximin value, regardless of how the others are playing.

The player doesn’t have the know the payoffs of the other players in order to choose the maximin move, therefore players can choose the maximin strategy in a simultaneous game regardless of what the other players choose.