## Introductory Game Theory: Most Common Types of Games

I recently came across another game theory book, Essentials of Game Theorythat introduces the fundamental math concepts behind the discipline. Since I’ve been doing quite a bit of coding over the last few months, I thought it might be a good idea to refresh myself on the math concepts in game theory. Hope this also helps anyone else who wants to learn/relearn the different types of games listed below.

1. Normal-form game: The most common game in game theory is the normal-form game where players with different strategies compete with each other for utility maximization. In more mathematical terms, N is a finite set of players where each player is denoted by i. Each player, $n_i$, has a set of actions available: $A_i$. For each game, every player may take an action, and the set of all players’ strategies is the vector $a$ where $a = A_1 X ... X A_n$. The utility obtained is denoted by $u = (u_1, ..., u_n)$ where $u_i$ is the utility received by a player for a particular action $A_i -> R$ for player $i$.
2. Common-payoff game: A game in which players are not competing with each other, but are coordinating on a set of actions that is mutually beneficial for all players. In other words, all the players in a game receive the same utility. Mathematically, given a set of possible actions, $a \epsilon A_1 X ... X A_n$ taken by players $i$ and $j$ (where $i, j \epsilon N$), the utility obtained by both players are the same, $u_i(a) = u_j(a)$.
3. Constant-sum game: This is a subset of the normal-form game in which the utility obtained by all players add to a constant sum. When one player gains in a constant sum game, the other player loses. In other words, in a two-player game, given the set of combinations of possible actions taken by each player $a \epsilon A_1 X A_2$ where $A_i$ is the set of strategies available to player $i$: the utility for each player obtained from the constant-sum game follows the rule: $u_1(a) + u_2(a) = c$. In a zero-sum game, $c = 0$. (Note: most constant-sum games you will encounter will be zero-sum games.)

I will be going over definitions for game theory strategies later on this week.