Game Theory Through Examples: Theory 2: Simultaneous Games
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Abstract
In his story “Jewish Poker” the writer Ephraim Kishon describes how a man called Ervinke convinces the narrator to play a game called Jewish Poker with him. “You think of a number, I also think of a number”, Ervinke explains. “Whoever thinks of a higher number wins. This sounds easy, but it has a hundred pitfalls.” Then they play. It takes the narrator some time until he realizes that it is better to let Ervinke tell his number first. [ K1961 ] Obviously this is a game that is not fair unless both players play simultaneously. In this chapter we will start our journey through game theory by considering games where each player moves only once, and moves are made simultaneously. The games can be described in a table (called the game's normal form ). Then we discuss approaches that allow the players to decide which move they will choose, culminating with the famous Nash equilibrium. Normal Form—Bimatrix Description Imagine you want to describe a simultaneous game. We know that each player has only one move, and that all moves are made simultaneously. What else do we need to say? First, we must stipulate the number of players in the game. Second, we must list for each player all possible moves. Different players may have different roles and may have different options for moves. We assume that each player has only finitely many options. Players simultaneously make their moves, determine the outcome of the game, and receive their payoffs. We need to describe the payoffs for each outcome.