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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.