Simultaneous game
Simultaneous game
thumb|upright=1.2|Rock–paper–scissors is an example of a simultaneous game.
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 exactly the same time. The security level for Player i is the amount max min Hi (s) that the player can guarantee themselves unilaterally, that is, without considering the actions of other players.
Representation In a simultaneous game, players will make their moves simultaneously, determine the outcome of the game and receive their payoffs.
The most common representation of a simultaneous game is normal form (matrix form). For a 2 player game; one player selects a row and the other player selects a column at exactly the same time. Traditionally, within a cell, the first entry is the payoff of the row player, the second entry is the payoff of the column player. The “cell” that is chosen is the outcome of the game. due to the party being unaware of the other party's decision (by definition of "simultaneous game"). [[File:Simultaneous_game.png|alt=|center|frame| The simultaneous game of rock–paper–scissors modeled in extensive form
Bimatrix game In a simultaneous game, players only have one move and all players' moves are made simultaneously. The number of players in a game must be stipulated and all possible moves for each player must be listed. Each player may have different roles and options for moves. However, each player has a finite number of options available to choose.
Two players An example of a simultaneous 2-player game:
A town has two companies, A and B, who currently make $8,000,000 each and need to determine whether they should advertise. The table below shows the payoff patterns; the rows are options of A and the columns are options of B. The entries are payoffs for A and B, respectively, separated by a comma.
Pure vs mixed strategy Pure strategies are those in which players pick only one strategy from their best response. A Pure Strategy determines all your possible moves in a game, it is a complete plan for a player in a given game. Mixed strategies are those in which players randomize strategies in their best responses set. These have associated probabilities with each set of strategies.
When analyzing a simultaneous game: * 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. Therefore, resulting in the following payoff matrix:
This game results in a clear dominant strategy of betrayal where the only strong Nash Equilibrium is for both prisoners to confess. This is because we assume both prisoners to be rational and possessing no loyalty towards one another. Therefore, betrayal provides a greater reward for a majority of the potential outcomes.
The stag hunt thumb|[[stag-hunt]]
The stag-hunt by philosopher Jean-Jacques Rousseau is a simultaneous game in which there are two players. The decision to be made is whether or not each player wishes to hunt a stag or a hare. Naturally hunting a stag will provide greater utility in comparison to hunting a hare. However, in order to hunt a stag both players need to work together. On the other hand, each player is perfectly capable of hunting a hare alone. The resulting dilemma is that neither player can be sure of what the other will choose to do. Thus, providing the potential for a player to receive no payoff should they be the only party to choose to hunt a stag. Therefore, resulting in the following payoff matrix:
The game is designed to illustrate a clear Pareto optimality where both players cooperate to hunt a Stag. However, due to the inherent risk of the game, such an outcome does not always come to fruition. It is imperative to note that Pareto optimality is not a strategic solution for simultaneous games. However, the ideal informs players about the potential for more efficient outcomes. Moreover, potentially providing insight into how players should learn to play over time.