DominanceOverviewAn Example – Prisoner’s DilemmaSlide 4RationalitySolving Using DominanceTeam ProductionCosts of EffortOptimal EffortEquilibrium EffortEquilibrium Effort ContinuedSolving the Free Rider ProblemBonusesEquilibrium AnalysisDesign ChoicesIterative EliminationDominance Solvable GamesMore on Dominance SolutionsWeak DominanceAuctionsA Model of eBayBidder’s ProblemGraphically – Bid ShadingGraphically – Bidding Above ValueGraphically – Bid = ValueCommentsSlide 39Case Study: Tender OffersTenders...DetailsDominance of the tenderA White KnightDominance revisitedSummaryDominanceOverviewIn this unit, we explore the notion of dominant strategiesDominance often requires weaker views of rationality than does standard equilibrium playThese weaker rationality requirements support choice of equilibria satisfying dominance over other equilibria.An Example – Prisoner’s DilemmaCooperate DefectCooperate3, 3 0, 4Defect4, 0 1, 1In this game, it pays to defect regardless of the rival’s strategyDefect is a best response to cooperateDefect is a best response to defectAn Example – Prisoner’s DilemmaCooperate DefectCooperate3, 3 0, 4Defect4, 0 1, 1In the language of dominance:The cooperate strategy is strictly dominated by defectThis means that the defect strategy gives strictly higher payoffs for Rowena than does cooperateRationalityRationality axiom 1: Never play a strictly dominated strategy regardless of your opponentWhy?Even if you have serious doubts about the rationality of the other player.A dominated strategy does strictly worse than some other strategy…regardless of your rival’s playSo it should be avoided.Solving Using DominanceCooperate DefectCooperate3, 3 0, 4Defect4, 0 1, 1In the prisoner’s dilemma, we can solve the game purely by eliminating dominated strategiesSince this elimination leaves each side only one undominated strategy, this pair constitutes an equilibrium.Team ProductionBoth the design and the production departments are required to produce some saleable output.The quality of the output determines the price for which it can be sold.For each unit of effort undertaken by either team, up to 10 units, profits increase by $1.5million/unit. After that, it does not increase.Costs of EffortIt costs $1million per unit of effort in either departmentEffort is unobservable by managementTo compensate design and production, management has instituted a profit sharing plan whereby production and design each get one-third of the profits as compensation.Optimal EffortFrom the perspective of the firm as a whole, each unit of effort up to 10 taken by design and production costs only $1m and has a return of 50%Therefore from the firm’s perspective each department should exert 10 units of effortEquilibrium EffortNotice that the design team needs to determine its level of effort not knowing the choice of the production team.What are its profits if design chooses effort e1 and production chooses e2?Profit1 = Profit share – Cost of effortProfit1 = (1/3)(1.5e1 + 1.5e2) – e1Equilibrium Effort ContinuedProfit1 = (1/3)(1.5e1 + 1.5e2) – e1Notice that regardless of e2, Profit1 is decreasing in e1So any choice e1 > 0 is dominated by e1 = 0.Hence design exerts no special effort despite the profit sharing incentivesThe situation for production is analogousThe conclusion is that both production and design will try to free ride off the efforts of the other and no effort will occurSolving the Free Rider ProblemFree rider problems appear in numerous settingsDevising incentive schemes to solve these problems is criticalWhat was wrong with the profit sharing scheme?BonusesSuppose that instead of doing a straight profit sharing arrangement, the firm uses a bonus system to compensate design and production.Recall that if production were efficient, profits would be $30m and the profit share gave away 2/3rds of this amount or $20m.Instead, suppose that the firm pays each team a bonus of $10m + $1 if they reach the profit target of $30m.Equilibrium AnalysisSuppose that design expects production to work all-out to meet the target. To receive the bonus, design has to work all-out too.If it doesn’t, then the analysis is as it was before but without even the profit sharing incentive---therefore design either works all-out or not at all.How do these situations compare?Design ChoicesIf design doesn’t work, it earns zeroIf they works all-out, profits equal the bonus less the cost of effort, which nets design $1.Thus, it is better to work all-out than not at all, so a best response to production’s working all-out is for design to do likewiseBottom line: The structure of incentive schemes (as well as the total amount) can have a big effect on free-rider problems.Iterative EliminationRecall that rationality axiom #1 prescribed that it was never a good idea to play a dominated strategyIf you have some confidence of your rival’s rationality, you might be willing to assume that she follows this axiom as well. This suggests that you should eliminate her dominated strategies in thinking about the game.Dominance Solvable GamesTo use dominance to solve a game:Delete dominated strategies for each of the playersLook at the smaller game with these strategies eliminatedNow delete dominated strategies for each side from the smaller gameContinue this process until no further deletion is possibleIf only single strategies remain, the game is dominance solvableMore on Dominance SolutionsNot all games are dominance solvableIf after elimination, a small set of strategies remain for each playerThese strategies survive iterative dominance and are relatively more robust than othersWeak DominanceTo eliminate a strategy as being dominated, we required that some other strategy always be better no matter the rival’s moveSuppose we weaken this:A strategy is weakly dominated if, no matter what the rival does, there is some strategy that does equally well and sometimes strictly better.AuctionseBay and a number of other online auctions use “proxy bidding” rulesUnder a proxy bid, you enter a bid amount, but what you pay is determined by the second highest bid plus a small increment. Suppose that you know your willingness to pay for an item for sale on eBay. What should you bid?A Model of