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## Games with Simultaneous Moves

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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 – Variable Sum Game

- Consider the negotiation game described earlier
- Row chooses between demanding small, medium, and large shares
- As does column

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.

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

For row: High is a best response to Low

Game Table – Column Analysis

For column: High is a best response to Low

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

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

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.

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