18 Simultaneous Games

A game is an interactive situation involving multiple self-interested agents, called players. The defining characteristic of a game is interdependence: each player’s outcome depends not only on their own choice but also on the choices of all other players.

Every game has three essential components:

Players: A finite set of decision-makers, denoted by $N$.
Actions/Strategies: For each player $i \in N$, a finite set $S_i$ of feasible options.
Utilities: For each player $i\in N$, a utility function $u_i$ representing their preferences over game outcomes, where outcomes are determined by the combination of all players’ actions.

Game theory includes many types of games, including sequential games where players move in order and stochastic games with random events. This course focuses on simultaneous move games (also called strategic form or normal form games), where all players choose their actions at the same time without observing others’ choices.

Definition 18.1 (Simultaneous Move Game) A game is a tuple $\langle N , (S_i)_{i\in N}, (u_i)_{i\in N }\rangle$ where

$N$ is a nonempty finite set of players,
for each $i\in N$, $S_i$ is a finite nonempty set of actions for player $i$, and
for each $i\in N$, $u_i:\times_{k\in N } S_k\rightarrow\mathbb{R}$ is player $i$’s utility function.

Here, $\times_{k\in N} S_k$ is the Cartesian product of the action sets, containing all possible combinations of actions (one per player).

To illustrate this definition, consider the folllowing two-player game:

$N = \{Row, Column\}$
$S_{Row} = \{u, d\}$, $S_{Column} = \{l, r\}$
$u_{Row}(u,l) = 2$, $u_{Row}(u,r) = 0$, $u_{Row}(d,l) = 0$, $u_{Row}(d,r) = 1$
$u_{Column}(u,l) = 1$, $u_{Column}(u,r) = 0$, $u_{Column}(d,l) = 0$, $u_{Column}(d,r) = 2$

This game can be depicted as the following matrix, where the rows correspond to Row’s actions and the columns correspond to Column’s actions. Each cell shows the resulting payoffs as an ordered pair, with Row’s payoff listed first and Column’s second.

		Column
		$l$	$r$
Row	$u$	$2, 1$	$0, 0$
Row	$d$	$0, 0$	$1, 2$

This matrix representation makes it easy to see all possible outcomes at a glance. For any combination of actions, you can immediately identify each player’s payoff—information that is essential for strategic analysis.

A strategy profile is a list of actions, one from each player. The set of all strategy profiles is $\times_{k\in N} S_k$. In the above example, the four strategy profiles are: \[\{(u,l), (u,r), (d,l), (d,r)\}.\]

Each strategy profile determines an outcome of the game, and each player has preferences over these outcomes expressed through their utility function. For instance, in the profile $(u,l)$, Row receives utility 2 while Column receives utility 1.

When analyzing simultaneous move games, remember that each player aims to maximize their own utility. These utility values capture all relevant aspects of a player’s preferences over outcomes—not just winning or losing, but any factor the player cares about (monetary payoffs, reputation, fairness, etc.).

18.1 Exercises

Consider a game with players $N = \{A, B\}$, where Player $A$ has actions $\{High, Low\}$ and Player $B$ has actions $\{Left, Right\}$.
1. How many strategy profiles does this game have?
  
  Show Answer
  
  There are $4$ strategy profiles: $$(High, Left), (High, Right), (Low, Left), (Low, Right)$
2. If we add a third player $C$ with actions $\{Up, Down\}$, how many strategy profiles would the game have?
  
  Show Answer
  
  There are $8$ strategy profiles. Each of the original $4$ profiles can be combined with either Up or Down, giving $4 \times 2 = 8$ total.
Consider the following game:

Player 2

$X$

$Y$

Player 1

$A$

$3, 3$

$0, 5$

$B$

$5, 0$

$1, 1$
1. What is Player 1’s utility when the strategy profile is (A, Y)?
  
  Show Answer
  
  When the profile is $(A, Y)$, Player 1 gets utility $0$ (the first number in the cell for row $A$, column $Y$).
2. Does Player 2 prefer the outcome $(B, X)$ to $(A, X)$?
  
  Show Answer
  
  At $(B, X)$, Player 2 gets $0$. At $(A, X)$, Player 2 gets $$3$. Since $3 > 0$, Player 2 prefers $(A, X)$.
3. List all strategy profiles where both players receive the same utility.
  
  Show Answer
  
  Two profiles: $(A, X)$ where both get $3$, and $(B, Y)$ where both get $1$.

		Column
		\(l\)	\(r\)
Row	\(u\)	\(2, 1\)	\(0, 0\)
Row	\(d\)	\(0, 0\)	\(1, 2\)

		Player 2
		\(X\)	\(Y\)
Player 1	\(A\)	\(3, 3\)	\(0, 5\)
Player 1	\(B\)	\(5, 0\)	\(1, 1\)