Presentation on theme: "Games with Simultaneous Moves"— Presentation transcript:
1 Games with Simultaneous Moves Nash equilibrium and normal form games
2 OverviewIn many situations, you will have to determine your strategy without knowledge of what your rival is doing at the same timeProduct designPricing and marketing some new productMergers and acquisitions competitionVoting and politicsEven 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 sumPure allocation of fixed surplusVariable SumSurplus is variable as is its allocation
4 Constant sum gamesSuppose that the “pie” is of fixed size and your strategy determines only the portion you will receive.These games are constant sum gamesCan always normalize the payoffs to sum to zeroPurely distributive bargaining and negotiation situations are classic examplesExample: Suppose that you are competing with a rival purely for market share.
5 Variable Sum GamesIn many situations, the size and the distribution of the pie are affected by strategiesThese games are called variable sumBargaining situations with both an integrative and distributive component are examples of variable sum gamesExample: 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 GameThis 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 strategiesThe 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 campaignRow firm can choose from among:Standard, medium risk, paradigm shiftColumn can choose among:Defend against standard, defend against medium, defend against paradigm shift
9 Game Table – Payoffs Defend Standard Defend Medium Defend Paradigm 20%50%80%Medium Risk60%56%70%Paradigm Shift90%40%10%
10 Game Table – Variable Sum Game Consider the negotiation game described earlierRow chooses between demanding small, medium, and large sharesAs does column
11 Game Table – Payoffs Low Medium High 25, 25 25, 50 25, 75 50, 25 50, 500, 075, 25
12 Solving Game TablesTo “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 payoffGiven this, choose the strategy where the minimum payoff is highest.That is, maximize the amount of the minimum payoffThis is called a maximin strategy.
15 Constant Sum Game – Finding Equilibrium Defend StandardDefend MediumDefend ParadigmMinStandard20%50%80%Medium Risk60%56%70%Paradigm Shift90%40%10%Max
16 Constant Sum Game – Row’s Best Strategy Defend StandardDefend MediumDefend ParadigmMinStandard20%50%80%Medium Risk60%56%70%Paradigm Shift90%40%10%Max
17 Constant Sum Game – Column’s Best Strategy Defend StandardDefend MediumDefend ParadigmMinStandard20%50%80%Medium Risk60%56%70%Paradigm Shift90%40%10%Max
18 Constant Sum Game – Equilibrium Defend StandardDefend MediumDefend ParadigmMinStandard20%50%80%Medium Risk60%56%70%Paradigm Shift90%40%10%Max
19 CommentsUsing 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 mediumWhen row plays medium, column can do no better than to defend against it.The strategies form mutual best responsesHence, we have found an equilibrium.
20 CaveatsMaximin 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 tableBut it’s time consuming for more complicated games.
22 Game Table – Row Analysis LowMediumHigh25, 2525, 5025, 7550, 2550, 500, 075, 25For row: High is a best response to Low
23 Game Table – Row’s Best Responses LowMediumHigh25, 2525, 5025, 7550, 2550, 500, 075, 25
24 Game Table – Column Analysis LowMediumHigh25, 2525, 5025, 7550, 2550, 500, 075, 25For column: High is a best response to Low
25 Game Table – Column’s Best Responses LowMediumHigh25, 2525, 5025, 7550, 2550, 500, 075, 25
27 Summary In this game, there are three pairs of mutual best responses The parties coordinate on an allocation of the pie without excess demandsBut any allocation is an equilibrium
28 Strategies that are continuous variables In many situations, it makes sense to model the strategies as being continuous rather than coming from a small set of choicesClassic example: competition in quantities (Cournot competition)
29 Cournot CompetitionSuppose that two firms are competing by choosing quantities of goods to place on the marketBoth have identical, constant marginal costsNormalize these to be zeroEveryone faces a linear demand curve
30 MonopolyIf firm 1 anticipates that firm 2 will choose to produce NO output, then firm 1 is a monopolist in this marketWe can find firm 1’s best response to this conjecture via the usual solution to monopoly problems
32 Comments Notice that marginal revenue lies below demand curve Discounts to attract the next customer have to be passed along to all the other existing customersTherefore, the increment to revenue is less than the willingness to pay of the marginal customer
33 Graphical Solution – Part 2 PriceDemandPmMRQmQuantity
34 Other ConjecturesNow suppose that firm 1 conjectures that firm 2 will produce 10 units.Firm 1 faces the following demand curvePQ10
35 CommentsNotice that this just shifts the location of the y-axis in the standard diagramThe “best response” of firm 1 is calculated the same way---solve the monopoly problem for the “adjusted” demand curve.
38 Some MathWe can write the graphical situation down as an optimization problem.Firm 1 conjectures firm 2’s quantity, q2, and chooses its own quantity, q1, to maximize profitsSuppose demand is P = 12 – Q, thenChoose q1 to maximizeProfit1 = q1 x PProfit1 = q1 x (12 – q1 – q2)
39 More Geometric Intuition Profit1 = q1 x (12 – q1 – q2)Notice that if q1 = 0, firm 1 earns no profitsLikewise if q1 = 12 – q2Thus, the profit function looks likeSlope is flat atThe topProfitq1
40 Calculus Profit1 = q1 x (12 – q1 – q2) Differentiate with respect to q1 and find the q1 at the top of the hill (i.e. slope = 0)This yields:d Profit1/dq1 = 12 – 2q1 – q2(This is the slope of the hill)12 – 2q1 – q2 = 0(This is the top of the hill – slope = 0)q1 = q2(This is the best response function)
41 Equilibrium An equilibrium is a pair of mutual best responses. This means that each side conjectures the other’s move correctly and best responds to it.So we need q1 and q2 solvingq1 = q2q2 = q1Solving: q1 = q2 = 4
42 Sequential Competition Now let’s compare this to the situation where firm 1 moves first --- and is observed --- followed by firm 2.Firm 2’s problem is just the same as it was before.Therefore, firm 1 anticipates that if it chooses q1, firm 2 will chooseq2 = q1Knowing this, what should 1 choose?
43 Firm 1’s OptimizationFirm 1 should look forward and reason back in making its decisionRecall:Profit1 = q1 x (12 – q1 – q2)But firm 1 knows (looking forward) thatq2 = q1Therefore, firm 1 will choose q1 to maximizeProfit1 = q1 x (12 – q1 –(6 - .5q1))Profit1 = q1 x (6 – .5q1)
44 CalculusOnce again, firm 1 seeks to get to the top of its profit hill:Profit1 = q1 x (6 – .5q1)dProfit1/dq1 = 6 – q1The slope is flat at the top6 – q1 = 0q1 = 6And, knowing q1, q2 = 3
45 CommentsGoing first in this game (Stackelberg competition) enables firm 1 to gain market share at firm 2’s expenseEven though the market price is now lower…(8 units of the good on the market when moves were simultaneous versus 9 now)Firm 1’s profits are higherHow do we know this without calculating it?This game has a first-mover advantage
46 How did firm 1 gain an advantage? CommitmentFirm 1 could commit to produce more when its production decision was observable by 2Strategic substitutesFurther, firm 1’s good is a (perfect) substitute for 2’s good.By committing to produce more, firm 2 was obliged to scale back productionThus, the two goods are strategic substitutes.
47 Other Archetypal Strategic Situations We close this unit by briefly studying some other common strategic situations
48 Hawk-DoveIn this situation, the players can either choose aggressive (hawk) or accommodating strategiesFrom each players perspective, preferences can be ordered from best to worst:Hawk – DoveDove – DoveDove – HawkHawk – HawkThe argument here is that two aggressive players wipe out all surplus
49 Hawk-Dove Analysis We can draw the game table as: Best Responses: Reply Dove to HawkReply Hawk to DoveEquilibriumThere are two equilibriaHawk-DoveDove-HawkHawkDove0, 04, 11, 42, 2
50 Battle of the SexesIn this game, surplus is obtained only if we agree to an actionHowever, the players differ in their opinions about the preferred actionAll surplus is lost if no agreement is reachedThere are two strategies: Value or Cost
51 PayoffsSuppose that the column player prefers the cost strategy and row prefers the value strategyPreference ordering for Row:Value-ValueCost-CostAnything elsePreference ordering for Column
52 BoS Analysis We can draw the game table as: Best Responses: Reply Value to ValueReply Cost to CostEquilibriumThere are two equilibriaValue-ValueCost-CostValueCost2, 10, 01, 2
53 Conclusions This is called Nash equilibrium Simultaneous games are those where your opponent’s strategy choice is unknown at the time you choose a strategyTo solve a simultaneous game, we look for mutual best responsesThis is called Nash equilibriumDrawing a game table is a useful way to analyze these types of situationsWhen there are many strategies, using best-response analysis can help to determine proper strategyGames may have several equilibria.Focal points and framing effects to steer the negotiation to the preferred equilibrium.