Games with Simultaneous Moves

Nash equilibrium and normal form games

Overview

In many situations, you will have to determine your strategy without knowledge of what your rival is doing at the same time Product design Pricing and marketing some new product Mergers and acquisitions competition Voting and politics

Even if the moves are not literally taking place at the same moment, if your move is in ignorance of your rival’s, the game is a simultaneous game

Two classes of Simultaneous Games

Constant sum Pure allocation of fixed surplus

Variable Sum Surplus is variable as is its allocation

Constant sum games

Suppose that the “pie” is of fixed size and your strategy determines only the portion you will receive.

These games are constant sum games Can always normalize the payoffs to sum to zero Purely distributive bargaining and negotiation

situations are classic examples Example: Suppose that you are competing

with a rival purely for market share.

Variable Sum Games

In many situations, the size and the distribution of the pie are affected by strategies

These games are called variable sum Bargaining situations with both an integrative and

distributive component are examples of variable sum games

Example: Suppose that you are in a negotiation with another party over the allocation of resources. Each of you makes demands regarding the size of the pie.

In the event that the demands exceed the total pie, there is an impasse, which is costly.

Nash Demand Game

This bargaining game is called the Nash demand game.

Constructing a Game Table

In simultaneous move games, it is sometimes useful to construct a game table instead of a game tree.

Each row (column) of the table corresponds to one of the strategies

The cells of the table depict the payoffs for the row and column player respectively.

Game Table – Constant Sum Game

Consider the market share game described earlier.

Firms choose marketing strategies for the coming campaign

Row firm can choose from among: Standard, medium risk, paradigm shift Column can choose among: Defend against standard, defend against

medium, defend against paradigm shift

Game Table – Payoffs

Defend Standard

Defend Medium

Defend Paradigm

Standard 20% 50% 80%

Medium Risk

60% 56% 70%

Paradigm Shift

90% 40% 10%

Game Table – Variable Sum Game

Consider the negotiation game described earlier

Row chooses between demanding small, medium, and large shares

As does column

Game Table – Payoffs

Low Medium High

Low 25, 25 25, 50 25, 75

Medium 50, 25 50, 50 0, 0

High 75, 25 0, 0 0, 0

Solving Game Tables

To “solve” a game table, we will use the notion of Nash equilibrium.

Solving Game Tables

Terminology Row’s strategy A is a best response to

column’s strategy B if there is no strategy for row that leads to higher payoffs when column employs B.

A Nash equilibrium is a pair of strategies that are best responses to one another.

Finding Nash Equilibrium – Minimax method In a constant sum game, a simple way to find

a Nash equilibrium is as follows: Assume that your rival can perfectly forecast

your strategy and seeks to minimize your payoff

Given this, choose the strategy where the minimum payoff is highest.

That is, maximize the amount of the minimum payoff

This is called a maximin strategy.

Constant Sum Game – Finding Equilibrium

Defend Standard

Defend Medium

Defend Paradigm

Min

Standard 20% 50% 80% 20%

Medium Risk

60% 56% 70% 56%

Paradigm Shift

90% 40% 10% 10%

Max 90% 56% 80%

Constant Sum Game – Row’s Best Strategy

Defend Standard

Defend Medium

Defend Paradigm

Min

Standard 20% 50% 80% 20%

Medium Risk

60% 56% 70% 56%

Paradigm Shift

90% 40% 10% 10%

Max 90% 56% 80%

Constant Sum Game – Column’s Best Strategy

Defend Standard

Defend Medium

Defend Paradigm

Min

Standard 20% 50% 80% 20%

Medium Risk

60% 56% 70% 56%

Paradigm Shift

90% 40% 10% 10%

Max 90% 56% 80%

Constant Sum Game – Equilibrium

Defend Standard

Defend Medium

Defend Paradigm

Min

Standard 20% 50% 80% 20%

Medium Risk

60% 56% 70% 56%

Paradigm Shift

90% 40% 10% 10%

Max 90% 56% 80%

Comments

Using minimax (and maximin for column) we conclude that medium/defend medium is the equilibrium.

Notice that when column defends the medium strategy, row can do no better than to play medium

When row plays medium, column can do no better than to defend against it.

The strategies form mutual best responses Hence, we have found an equilibrium.

Caveats

Maximin analysis only works for zero or constant sum games

Finding an Equilibrium – Cell-by-Cell Inspection This is a low-tech method, but will work for all games. Method:

Check each cell in the matrix to see if either side has a profitable deviation.

A profitable deviation is where by changing his strategy (leaving the rival’s choice fixed) a player can improve his or her payoffs.

If not, the cell is a best response. Look for all pairs of best responses.

This method finds all equilibria for a given game table But it’s time consuming for more complicated games.

Game Table – Row Analysis

Low Medium High

Low 25, 25 25, 50 25, 75

Medium 50, 25 50, 50 0, 0

High 75, 25 0, 0 0, 0

For row: High is a best response to Low

Game Table – Row’s Best Responses

Low Medium High

Low 25, 25 25, 50 25, 75

Medium 50, 25 50, 50 0, 0

High 75, 25 0, 0 0, 0

Game Table – Column Analysis

Low Medium High

Low 25, 25 25, 50 25, 75

Medium 50, 25 50, 50 0, 0

High 75, 25 0, 0 0, 0

For column: High is a best response to Low

Game Table – Column’s Best Responses

Low Medium High

Low 25, 25 25, 50 25, 75

Medium 50, 25 50, 50 0, 0

High 75, 25 0, 0 0, 0

Game Table – Equilibrium

Low Medium High

Low 25, 25 25, 50 25, 75

Medium 50, 25 50, 50 0, 0

High 75, 25 0, 0 0, 0

Summary

In this game, there are three pairs of mutual best responses

The parties coordinate on an allocation of the pie without excess demands

But any allocation is an equilibrium

Other Archetypal Strategic Situations

We close this unit by briefly studying some other common strategic situations

Hawk-Dove

In this situation, the players can either choose aggressive (hawk) or accommodating strategies

From each players perspective, preferences can be ordered from best to worst: Hawk – Dove Dove – Dove Dove – Hawk Hawk – Hawk

The argument here is that two aggressive players wipe out all surplus

Hawk-Dove Analysis

We can draw the game table as:

Best Responses: Reply Dove to Hawk Reply Hawk to Dove

Equilibrium There are two

equilibria Hawk-Dove Dove-Hawk

Hawk Dove

Hawk 0, 0 4, 1

Dove 1, 4 2, 2

Battle of the Sexes

In this game, surplus is obtained only if we agree to an action

However, the players differ in their opinions about the preferred action

All surplus is lost if no agreement is reached There are two strategies: Value or Cost

Payoffs

Suppose that the column player prefers the cost strategy and row prefers the value strategy

Preference ordering for Row: Value-Value Cost-Cost Anything else

Preference ordering for Column Cost-Cost Value-Value Anything else

BoS Analysis

We can draw the game table as:

Best Responses: Reply Value to Value Reply Cost to Cost

Equilibrium There are two

equilibria Value-Value Cost-Cost

Value Cost

Value 2, 1 0, 0

Cost 0, 0 1, 2

Conclusions

Simultaneous games are those where your opponent’s strategy choice is unknown at the time you choose a strategy

To solve a simultaneous game, we look for mutual best responses This is called Nash equilibrium

Drawing a game table is a useful way to analyze these types of situations

When there are many strategies, using best-response analysis can help to determine proper strategy

Games may have several equilibria. Focal points and framing effects to steer the

negotiation to the preferred equilibrium.