Games with Simultaneous Moves

1. Games with Simultaneous Moves Nash equilibrium and normal form games

2. 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

3. Two classes of Simultaneous Games • Constant sum • Pure allocation of fixed surplus • Variable Sum • Surplus is variable as is its allocation

4. 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.

5. 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.

6. Nash Demand Game • This bargaining game is called the Nash demand game.

7. 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.

8. 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

9. Game Table – Payoffs

10. Game Table – Variable Sum Game • Consider the negotiation game described earlier • Row chooses between demanding small, medium, and large shares • As does column

11. Game Table – Payoffs

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

13. 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.

14. 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.

15. Constant Sum Game – Finding Equilibrium

16. Constant Sum Game – Row’s Best Strategy

17. Constant Sum Game – Column’s Best Strategy

18. Constant Sum Game – Equilibrium

19. 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.

20. Caveats • Maximin analysis only works for zero or constant sum games

21. 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.

22. Game Table – Row Analysis For row: High is a best response to Low

23. Game Table – Row’s Best Responses

24. Game Table – Column Analysis For column: High is a best response to Low

25. Game Table – Column’s Best Responses

26. Game Table – Equilibrium

27. 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

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

29. 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

30. 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

31. 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

32. 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

33. 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

34. 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.