Games with Simultaneous MovesOverviewTwo classes of Simultaneous GamesConstant sum gamesVariable Sum GamesNash Demand GameConstructing a Game TableGame Table – Constant Sum GameGame Table – PayoffsGame Table – Variable Sum GameSlide 11Solving Game TablesSlide 13Finding Nash Equilibrium – Minimax methodConstant Sum Game – Finding EquilibriumConstant Sum Game – Row’s Best StrategyConstant Sum Game – Column’s Best StrategyConstant Sum Game – EquilibriumCommentsCaveatsFinding an Equilibrium – Cell-by-Cell InspectionGame Table – Row AnalysisGame Table – Row’s Best ResponsesGame Table – Column AnalysisGame Table – Column’s Best ResponsesGame Table – EquilibriumSummaryOther Archetypal Strategic SituationsHawk-DoveHawk-Dove AnalysisBattle of the SexesPayoffsBoS AnalysisConclusionsGames with Simultaneous MovesNash equilibrium and normal form gamesOverview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 gameTwo classes of Simultaneous GamesConstant sumPure allocation of fixed surplusVariable SumSurplus is variable as is its allocationConstant 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 shiftGame Table – PayoffsDefend StandardDefend MediumDefend ParadigmStandard 20% 50% 80%Medium Risk60% 56% 70%Paradigm Shift90% 40% 10%Game Table – Variable Sum GameConsider the negotiation game described earlierRow chooses between demanding small, medium, and large sharesAs does columnGame Table – PayoffsLow Medium HighLow 25, 25 25, 50 25, 75Medium 50, 25 50, 50 0, 0High 75, 25 0, 0 0, 0Solving 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.Constant Sum Game – Finding EquilibriumDefend StandardDefend MediumDefend ParadigmMinStandard 20% 50% 80% 20%Medium Risk60% 56% 70% 56%Paradigm Shift90% 40% 10% 10%Max 90% 56% 80%Constant Sum Game – Row’s Best StrategyDefend StandardDefend MediumDefend ParadigmMinStandard 20% 50% 80% 20%Medium Risk60% 56% 70% 56%Paradigm Shift90% 40% 10% 10%Max 90% 56% 80%Constant Sum Game – Column’s Best StrategyDefend StandardDefend MediumDefend ParadigmMinStandard 20% 50% 80% 20%Medium Risk60% 56% 70% 56%Paradigm Shift90% 40% 10% 10%Max 90% 56% 80%Constant Sum Game – EquilibriumDefend StandardDefend MediumDefend ParadigmMinStandard 20% 50% 80% 20%Medium Risk60% 56% 70% 56%Paradigm Shift90% 40% 10% 10%Max 90% 56% 80%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 gamesFinding 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 AnalysisLow Medium HighLow 25, 25 25, 50 25, 75Medium 50, 25 50, 50 0, 0High 75, 25 0, 0 0, 0 For row: High is a best response to LowGame Table – Row’s Best ResponsesLow Medium HighLow 25, 25 25, 50 25, 75Medium 50, 25 50, 50 0, 0High 75, 25 0, 0 0, 0Game Table – Column AnalysisLow Medium HighLow 25, 25 25, 50 25, 75Medium 50, 25 50, 50 0, 0High 75, 25 0, 0 0, 0 For column: High is a best response to LowGame Table – Column’s Best ResponsesLow Medium HighLow 25, 25 25, 50 25, 75Medium 50, 25 50, 50 0, 0High 75, 25 0, 0 0, 0Game Table – EquilibriumLow Medium HighLow 25, 25 25, 50 25, 75Medium 50, 25 50, 50 0, 0High 75, 25 0, 0 0,