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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

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games with simultaneous moves

Games with Simultaneous Moves

Nash equilibrium and normal form games

overview
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
Two classes of Simultaneous Games
  • Constant sum
    • Pure allocation of fixed surplus
  • Variable Sum
    • Surplus is variable as is its allocation
constant sum games
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
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
Nash Demand Game
  • This bargaining game is called the Nash demand game.
constructing a game table
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
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
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
Solving Game Tables
  • To “solve” a game table, we will use the notion of Nash equilibrium.
solving game tables13
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
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
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
Caveats
  • Maximin analysis only works for zero or constant sum games
finding an equilibrium cell by cell inspection
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
Game Table – Row Analysis

For row: High is a best response to Low

game table column analysis
Game Table – Column Analysis

For column: High is a best response to Low

summary
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
Other Archetypal Strategic Situations
  • We close this unit by briefly studying some other common strategic situations
hawk dove
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
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
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
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
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
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.