2. Static two person games

In this section we will look at static games in normal form where two players only move once (and simultaneously). While this is very restrictive, it allows us to introduce concepts that can be applied more generally without introducing too much complexity.

We begin with various notions that might capture what we may mean by a solution to a game.

Definition 2.1

A strategy \(\underline{x}\in \Delta S_1\) strictly dominates a (pure) strategy \(s\in S_1\) if \[\mathbb E_1(\underline{x},t)>\mathbb E_1(s,t)\] for all \(t\in S_{2}\). A strategy \(\underline{x}\in \Delta S_1\) weakly dominates a strategy \(s\in S_1\) if \[\mathbb E_1(\underline{x},t)\geq \mathbb E_1(s,t)\] for all \(t\in S_{2}\) and there exists some \(t\in S_{2}\) where this inequality is strict.

There are similar definitions for player two. For example a strategy \(\underline{y}\in \Delta S_2\) strictly dominates a (pure) strategy \(t\in S_2\) if \[\mathbb E_2(s,\underline{y})>\mathbb E_2(s,t)\] for all \(s\in S_{1}\).

For a pure strategy to dominate another in a bimatrix game, we merely need to compare the matrix entries. For player one (with payoff matrix \(A\)) the strategy \(s\) weakly dominates the strategy \(t\) if \[a_{sj}\geq a_{tj}\] for all \(j\), with strict inequality for at least one \(j\). For player two (with payoff matrix \(B\)) the strategy \(s\) weakly dominates the strategy \(t\) if \[b_{js}\geq b_{jt}\] for all \(j\), with strict inequality for at least one \(j\).

The dominance principle states that a rational player should never play in a strictly dominated strategy.

There are some occasions when we can solve a game using iterated dominance: the iterated removal of strictly dominated strategies. Even when this does not work, it can still be used to try to simplify the problem.

Example 2.2

Consider the game

Strategy 1 Strategy 2 Strategy 3 Strategy 4
Strategy 1 \((1,1)\) \((2,0)\) \((2,2)\) \((0,1)\)
Strategy 2 \((0,3)\) \((1,5)\) \((4,4)\) \((3,1)\)
Strategy 3 \((2,4)\) \((3,6)\) \((3,0)\) \((2,2)\)

Example completed by hand in the lecture

The next example shows why we need to consider mixed as well as pure strategies.

Example 2.3

Consider the game

L R
U \((4,1)\) \((0,2)\)
M \((0,0)\) \((4,0)\)
D \((1,3)\) \((1,2)\)

Example completed by hand in the lecture

In general, player \(i\) may be able to dominate some pure strategy \(s'\in S_i\) with a mixed strategy of the form \[\lambda_1s_1+\cdots + \lambda_rs_r\] where \(s_1,\ldots s_r\in S_i\) and \(\lambda_j\geq 0\) for all \(1\leq j\leq r\) and \[\sum_{j=1}^r\lambda_j=1.\]

For simplicity we will only consider here the case \(r=2\), but the approach we will discuss does generalise.

Consider a strategy of the form \(\lambda_1 s_1 + \lambda_2 s_2\) as above. It is easy to see that the maximum value of this strategy against any pure strategy for the other player is at most the maximum of the values of \(s_1\) and \(s_2\) against the same pure strategy.

Thus a pair of pure strategies \(s_1\) and \(s_2\) can only form a mixed strategy which strictly dominates \(s'\) if for each pure strategy \(t\) for the other player, at least one of \(s_1\) or \(s_2\) has a strictly greater value against \(t\) than \(s'\) does.

This condition is necessary but not sufficient, and so having identified a possible pair we need to determine whether a particular mixed strategy exists that dominates \(s'\). We illustrate this process with an example.

Example 2.4

Consider the game

P Q R
X \((5,1)\) \((1,2)\) \((3,6)\)
Y \((2,3)\) \((2,4)\) \((1,1)\)
Z \((1,5)\) \((4,2)\) \((0,3)\)

Example completed by hand in the lecture

Using the same method for columns instead of rows (and considering the values for player 2 instead of player 1) we can look for mixed strategies for player 2 that dominate one of their given pure strategies.

Sometimes we will be able to use iterated dominance to reduce a game to a single pair of pure strategies as in Example 2.2.

However, more typically we will not be able to solve a game using dominance (although it may still be useful for simplifying the problem).

More importantly, we do not yet have a notion of what it would mean to find a solution for a general game.

If player 1 has picked a given strategy, and player 2 can deduce that they have done so, will they not pick a strategy that counteracts this?

And given that player 1 will know that they are planning this, will they not in turn want to change their strategy in response?

It looks like we will end up going round in circles, with each player changing their choice of strategy in response to the other.

In the next lecture we shall see the key definition in this module, which resolves this problem and allows for a meaningful analysis of games.