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. A simple example is rock-paper-scissors in which all players make their choice at exactly the 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. Given that decision makers are rational, then so is individual rationality. An outcome is individually rational if it yields each player at least his security level. 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. To determine which "cell" is chosen, the payoffs for both the row player and the column player must be compared respectively. Each player is best off where their payoff is higher. Rock–paper–scissors, a widely played hand game, is an example of a simultaneous game. Both players make a decision without knowledge of the opponent's decision, and reveal their hands at the same time. There are two players in this game and each of them has three different strategies to make their decision; the combination of strategy profiles (a complete set of each player's possible strategies) forms a 3×3 table. We will display Player 1's strategies as rows and Player 2's strategies as columns. In the table, the numbers in red represent the payoff to Player 1, the numbers in blue represent the payoff to Player 2. Hence, the pay off for a 2 player game in rock-paper-scissors will look like this:

Another common representation of a simultaneous game is extensive form (game tree). Information sets are used to emphasize the imperfect information. Although it is not simple, it is easier to use game trees for games with more than 2 players. Even though simultaneous games are typically represented in normal form, they can be represented using extensive form too. While in extensive form one player’s decision must be draw before that of the other, by definition such representation does not correspond to the real life timing of the players’ decisions in a simultaneous game. The key to modeling simultaneous games in the extensive form is to get the information sets right. A dashed line between nodes in extensive form representation of a game represents information asymmetry and specifies that, during the game, a party cannot distinguish between the nodes, due to the party being unaware of the other party's decision (by definition of "simultaneous game").

Some variants of chess that belong to this class of games include synchronous chess and parity chess.

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.

Two players (zero sum) A zero-sum game is when the sum of payoffs equals zero for any outcome i.e. the losers pay for the winners gains. For a zero-sum 2-player game the payoff of player A doesn’t have to be displayed since it is the negative of the payoff of player B. An example of a simultaneous zero-sum 2-player game: Rock–paper–scissors is being played by two friends, A and B for $10. The first cell stands for a payoff of 0 for both players. The second cell is a payoff of 10 for A which has to be paid by B, therefore a payoff of -10 for B.

Three or more players An example of a simultaneous 3-player game: A classroom vote is held as to whether or not they should have an increased amount of free time. Player A selects the matrix, player B selects the row, and player C selects the column. The payoffs are:

Symmetric games All of the above examples have been symmetric. All players have the same options so if players interchange their moves, they also interchange their payoffs. By design, symmetric games are fair in which every player is given the same chances.

Strategies - the best choice Game theory should provide players with advice on how to find which move is best. These are known as “Best Response” strategies.

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