While the extensive form is a useful way to visualise the steps in a game, we will find it more convenient to work with the normal form instead. For this we will need some additional definitions.

### Definition 1.2

A strategy is a complete contingent plan for a player in a game. That is, a plan describing what move the player should make in each of their possible information sets.

The set of possible strategies for a given player is called the strategy set for that player.

Given a player $$i$$ we will let $$S_i$$ denote the strategy set for that player, and $$s_i\in S_i$$ will denote a particular strategy. A strategy profile is a vector of strategies, one for each player of the game, ie a vector $\underline{s}=(s_1,\ldots,s_n)$ where $$s_i\in S_i$$ for $$1\leq i\leq n$$.

We will let $$S$$ denote the set of all strategy profiles, so $S=S_1\times S_2\times\cdots \times S_n.$

For each player $$i$$ we can define a function $$u_i:S\longrightarrow \mathbb R$$ so that for each strategy profile $$\underline{s}\in S$$ the value of $$u_i(\underline{s})$$ represents the payoff to player $$i$$ in that game. The function $$u_i$$ is called the player’s payoff function.

### Definition 1.3

A game in normal form consists of a set $$\{1,\ldots,n\}$$ of players, corresponding strategy sets $$S_1,S_2,\ldots,S_n$$, and payoff functions $$u_1,u_2\ldots,u_n$$.

If we just have two players then we may suppose that player $$1$$ has $$p$$ strategies which we can number from $$1$$ to $$p$$ and player $$2$$ has $$q$$ strategies which we can number from $$1$$ to $$q$$.

Then a strategy profile is just a pair $$(x,y)$$, and we can write the payoff function for each player as a matrix where the $$(x,y)$$-entry is the value of the function for the given player for the strategy profile $$(x,y)$$.

Because each game can be described using a pair of matrices, such games are often called bimatrix games.

Here is an example of such a bimatrix game, where the payoff matrix for player 1 is $$A$$ and for player 2 is $$B$$.

Player 2
Strategy 1 Strategy 2 $$\cdots$$ Strategy $$q$$
Strategy 1 $$(a_{11},b_{11})$$ $$(a_{12},b_{12})$$ $$\cdots$$ $$(a_{1q},b_{1q})$$
Player 2 Strategy 2 $$(a_{21},b_{21})$$ $$(a_{22},b_{22})$$ $$\cdots$$ $$(a_{2q},b_{2q})$$
$$\vdots$$
Strategy $$p$$ $$(a_{p1},b_{p1})$$ $$(a_{p2},b_{p2})$$ $$\cdots$$ $$(a_{pq},b_{pq})$$

### Example 1.4

Construct the normal form for the game in Example 1.1.

Example completed by hand in the lecture

The normal form can be regarded as a model for situations where the players simultaneously and independently select strategies for an extensive form game. Such games are called one-shot or static games.

### Example 1.5: The Prisoners’ Dilemma

Example completed by hand in the lecture

### Example 1.6: Matching Pennies

Example completed by hand in the lecture

### Example 1.7: The Game of Chicken

Example completed by hand in the lecture

In some games you do not want to always pick the same strategy — for example in the rock, paper, scissors game. In such circumstance we will allow players to choose their strategies probabilistically.

### Definition 1.8

A mixed strategy for player $$i$$ is a vector $$\underline{x}=(x_1,\ldots,x_{r_i})$$ where $$r_i=|S_i|$$ and $$x_j\geq 0$$ for all $$1\leq j\leq r_i$$ and $\sum_{j=1}^{r_i}x_j=1.$

Here $$x_j$$ represents the probability that player i chooses strategy $$j$$.

The set $$\Delta S_i$$ of mixed strategies for player $$i$$ is defined by $\Delta S_i=\{(x_1,\ldots,x_{r_i}): \ x_j\geq 0\ \text{for all j and}\ \sum_{j=1}^{r_i}x_j=1\}.$ Thus a mixed strategy for player $$1$$ is an element of $$\Delta S_1$$ and a mixed strategy for player $$2$$ is an element of $$\Delta S_2$$.

If all of the $$x_j$$ except for one are equal to $$0$$ then we call this a pure strategy. These were the kinds of strategies we first considered.

If players use mixed strategies then we can only calculate the expected payoff from a game — ie the average payoff which they would receive if the game was repeated many times.

Suppose that player 1 has payoff matrix $$A$$ and uses the mixed strategy $$\underline{x}\in\Delta S_1$$, and player 2 has payoff matrix $$B$$, and uses $$\underline{y}\in\Delta S_2$$. We would like a formula for the expected payoff for each player.

The expected payoff for player 1 is given by $\begin{array}{rl} \mathbb E_1(\underline{x},\underline{y})=&\sum_{i=1}^{p}\sum_{j=1}^{q}a_{ij}\mathbb P(\text{1 uses strategy i and 2 uses strategy j})\\ =&\sum_{i=1}^{p}\sum_{j=1}^{q}a_{ij}\mathbb P(\text{1 uses strategy i})\mathbb P(\text{2 uses strategy j})\\ =&\sum_{i=1}^{p}\sum_{j=1}^{q}x_ia_{ij}y_j\\ =&\underline{x}A\underline{y}^T. \end{array}$ (Here we use the assumption that the players choose independently for the first step.)

Similarly player 2 has expected payoff $\mathbb E_2(\underline{x},\underline{y})=\underline{x}B\underline{y}^T.$

### Example 1.9

Suppose that in the prisoners’ dilemma the probability of player 1 confessing is 0.2 and of player 2 confessing is 0.4. Calculate the expected value for each player of the game.

Example completed by hand in the lecture