Games with Simultaneous Moves Nash equilibrium and normal

Games with Simultaneous Moves Nash equilibrium and normal form games

Overview n In many situations, you will have to determine your strategy without knowledge of what your rival is doing at the same time n n Product design Pricing and marketing some new product Mergers and acquisitions competition Voting and politics n 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 n Constant sum n Pure allocation of fixed surplus n Variable Sum n Surplus is variable as is its allocation

Constant sum games n Suppose that the “pie” is of fixed size and your strategy determines only the portion you will receive. n n 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 n n n In many situations, the size and the distribution of the pie are affected by strategies These games are called variable sum n 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. n In the event that the demands exceed the total pie, there is an impasse, which is costly.

Nash Demand Game n This bargaining game is called the Nash demand game.

Constructing a Game Table n In simultaneous move games, it is sometimes useful to construct a game table instead of a game tree. n Each row (column) of the table corresponds to one of the strategies n The cells of the table depict the payoffs for the row and column player respectively.

Game Table – Constant Sum Game n Consider the market share game described earlier. n Firms choose marketing strategies for the coming campaign n Row firm can choose from among: Standard, medium risk, paradigm shift n Column can choose among: n Defend against standard, defend against medium, defend against paradigm shift n

Game Table – Payoffs Defend Standard Medium Defend Paradigm Standard 20% 50% 80% Medium Risk 60% 56% 70% Paradigm 90% Shift 40% 10%

Game Table – Variable Sum Game n Consider the negotiation game described earlier n Row chooses between demanding small, medium, and large shares n 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

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

Solving Game Tables n Terminology n 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. n A Nash equilibrium is a pair of strategies that are best responses to one another.

Finding Nash Equilibrium – Minimax method n 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 n Given this, choose the strategy where the minimum payoff is highest. n That is, maximize the amount of the minimum payoff n This is called a maximin strategy. n

Constant Sum Game – Finding Equilibrium Defend Standard Defend Medium Defend Min Paradigm Standard 20% 50% 80% 20% Medium Risk 60% 56% 70% 56% Paradigm 90% Shift 40% 10% Max 56% 80% 90%

Constant Sum Game – Row’s Best Strategy Defend Standard Defend Medium Defend Min Paradigm Standard 20% 50% 80% 20% Medium Risk 60% 56% 70% 56% Paradigm 90% Shift 40% 10% Max 56% 80% 90%

Constant Sum Game – Column’s Best Strategy Defend Standard Defend Medium Defend Min Paradigm Standard 20% 50% 80% 20% Medium Risk 60% 56% 70% 56% Paradigm 90% Shift 40% 10% Max 56% 80% 90%

Constant Sum Game – Equilibrium Defend Standard Defend Medium Defend Min Paradigm Standard 20% 50% 80% 20% Medium Risk 60% 56% 70% 56% Paradigm 90% Shift 40% 10% Max 56% 80% 90%

Comments n Using minimax (and maximin for column) we conclude that medium/defend medium is the equilibrium. n Notice that when column defends the medium strategy, row can do no better than to play medium n When row plays medium, column can do no better than to defend against it. n The strategies form mutual best responses n Hence, we have found an equilibrium.

Caveats n Maximin analysis only works for zero or constant sum games

Finding an Equilibrium – Cell-by-Cell Inspection n This is a low-tech method, but will work for all games. n Method: n Check each cell in the matrix to see if either side has a profitable deviation. n A profitable deviation is where by changing his strategy (leaving the rival’s choice fixed) a player can improve his or her payoffs. n If not, the cell is a best response. n Look for all pairs of best responses. n This method finds all equilibria for a given game table n 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 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

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 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

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

Summary n In this game, there are three pairs of mutual best responses n The parties coordinate on an allocation of the pie without excess demands n But any allocation is an equilibrium

Other Archetypal Strategic Situations n We close this unit by briefly studying some other common strategic situations

Hawk-Dove n In this situation, the players can either choose aggressive (hawk) or accommodating strategies n From each players perspective, preferences can be ordered from best to worst: n n Hawk – Dove – Hawk n The argument here is that two aggressive players wipe out all surplus

Hawk-Dove Analysis Hawk Dove Hawk 0, 0 4, 1 Dove 1, 4 2, 2 n We can draw the game table as: n Best Responses: n n Reply Dove to Hawk Reply Hawk to Dove n Equilibrium n There are two equilibria n Hawk-Dove n Dove-Hawk

Battle of the Sexes n In this game, surplus is obtained only if we agree to an action n However, the players differ in their opinions about the preferred action n All surplus is lost if no agreement is reached n There are two strategies: Value or Cost

Payoffs n Suppose that the column player prefers the cost strategy and row prefers the value strategy n Preference ordering for Row: n n n Value-Value Cost-Cost Anything else n Preference ordering for Column n Cost-Cost n Value-Value n Anything else

Bo. S Analysis Value Cost Value 2, 1 0, 0 Cost 0, 0 1, 2 n We can draw the game table as: n Best Responses: n n Reply Value to Value Reply Cost to Cost n Equilibrium n There are two equilibria n Value-Value n Cost-Cost

Conclusions n Simultaneous games are those where your opponent’s strategy choice is unknown at the time you choose a strategy n To solve a simultaneous game, we look for mutual best responses n This is called Nash equilibrium n Drawing a game table is a useful way to analyze these types of situations n When there are many strategies, using best-response analysis can help to determine proper strategy n Games may have several equilibria. n Focal points and framing effects to steer the negotiation to the preferred equilibrium.