Equilibrium Pairs

Equilibrium Pairs#

Equilibrium Pairs

A pair of strategies is an equilibrium pair if neither player gains by changing strategies.

Thus, we see our textbook definition of an equilibrium is a point from which neither player wishes to deviate. In a contest where we repeatedly play this game, the players will settle on the equilibrium.

Constant Sum Games#

One special type of game can be solved exactly as zero sum games.

Constant Sum Games

A two player game is called a constant sum game if the sum of the payoffs to each player is constant for all possible outcomes of the game. More specifically, the coordinates in each payoff vector must add up to the same value for each payoff vector.

Note

Our textbook refers to constant sum games as zero sum games since the solution path for both is identical.

Cake Cutting Game#

As an example, our textbook points the following game where outcomes are shown in percentages of the cake.

\[\begin{split}\begin{array}{cc}&\text{Chooser}\\\text{Cutter}&\begin{array}{c|cc}&\text{Larger Piece}&\text{Smaller Piece}\\\hline\text{Cut Evenly}&(50,50)&(50,50)\\\text{Cut Unevenly}&(40,60)&(60,40)\end{array}\end{array}\end{split}\]

To divide the cake fairly between two children, say, one may cut the cake while the other may choose which of the two pieces to eat. The cutter knows that a completely unfair cut, say \((75,25)\), will result in the chooser getting much more cake.

Equilibria#

The chooser will always choose the larger piece if they are not even. Hence, the cutter does best to divide the cake evenly.

The equlibrium pair for this game is the strategy pair denoted \((\text{Cut Evenly}, \text{Larger Piece})\), with resulting payoff vector \((50,50)\). We will call this the outcome of the game or the equilibrium.

Finding Equilibrium Pairs#

We will develop mathematical strategies for finding the <span style=’color:blue;”>Nash Equilibrium of various matrix games. However, at this point in <span style=’color:blue;”>discovering game theory, the author of our textbook suggests the method of

guess and check

We proceed as follows:

  1. Assume Rose plays strategy A all of the time.

  2. What would Colin do?

  3. Assume Colin plays strategy A all of time.

  4. What would Rose do?

  5. What if Rose plays B or C?

  6. What if Colin plays B or C?

Repeating questions (1) and (2) for each pair of strategies, we may find that one pair is better for both Rose and Colin. If we do find such a pair, that is the equlibrium pair.

Warning

As we analyze, we must remember the rational play principle. Both Rose and Colin will play rationally and with self-interest. We must reason for each player as though they would choose the best strategy option to maximize their gains and minimize their losses.

At times, matrix games do not seem fair. When the equilibrium turns to be

\[(3, -3)\]

for example, we find that Rose is winning 3 units every time the game is played while Colin loses 3. While Colin is unhappy with the above example, Rose would be unhappy with the equilibrium below:

\[(-2,2)\]

This scenario is not uncommon in game theory. In fact, it happens often enough to earned a name.

Competitive Advantage

In a zero sum game where the equilbrium pair is \((a,-a)\) for some \(a > 0\), the value \(a\) is the competitive advantage for Player 1.

If \(a<0\), then we say that Player 2 would have a competitive advantage of \(|-a|\).

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