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Lecture 4B Nash Equilibrium. This lecture introduces the concept of Nash equilibrium. . Incentives in the Workplace. Consider a firm that sells output jointly produced by a design team and a manufacturing team. The quality of the output determines the price for which it can be sold.
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Lecture 4BNash Equilibrium This lecture introduces the concept of Nash equilibrium.
Incentives in the Workplace • Consider a firm that sells output jointly produced by a design team and a manufacturing team. • The quality of the output determines the price for which it can be sold. • To keep matters simple, we assume that for each extra unit of effort undertaken by either team, up to 10 units, sales rise by $1.5million per unit. • Any input beyond 10 within either team is wasted effort, producing no marginal increase in sales.
Cost of Effort • It costs $1million per unit of effort in either team, in terms of lost sleep, hiring new staff, and buying new equipment (plant) and materials. • Effort is not observed by the management. It is easy for each team to disguise sloth and inattention, thus confusing management about what needs to be done. The managers realize that.
The managers pitch their plan • To compensate design and manufacturing, management institutes a profit sharing plan, whereby production and manufacturing each get one-third of the sales as compensation. • Furthermore if the firm reaches its sales target of $30 million, it will also distribute a bonus of $100,000 dollars to both teams.
Potential benefits From the perspective of the firm as a whole, each unit of effort up to 10 taken by design and manufacturing costs $20 million and has a return of $30 million. If both teams follow the managers directives, they would earn a net $100,000 each. This way everyone in the firm benefits, management and shareholders most of all.
The design team chooses its effort • The design team determines its level of effort not knowing the choice of the production team. • What are its profits if the design team chooses effort e1 < 10 and manufacturing chooses e2? • Profit1 = Profit share – Cost of effort • Profit1 = (1/3)(1.5e1 + 1.5e2) – e1
Finding the effort level continued • Profit1 = (1/3)(1.5e1 + 1.5e2) – e1 • Regardless of e2, Profit1 is decreasing in e1. Therefore any choice with 10 > e1 > 0 is dominated by e1 = 0. • We only have to check whether both teams expending effort of 10 is rational. But if one team works that hard, the other team makes more profit by spending no effort at all. • Hence design exerts no special effort despite the profit sharing incentives. The situation for manufacturing is analogous. Spending no effort is a dominant strategy.
A large bonus only • Recall that if production and design were efficient, profits would be $30m and the profit share gave away 2/3rds of this amount or $20m plus the $200,00 bonus. • Instead, suppose the firm pays each team compensation of $10 million plus a $50,000 bonus each if they reach the profit target of $30m. Otherwise they get nothing. • This is less generous than the profit sharing scheme.
Analysis of the new incentive scheme • Suppose that design expects manufacturing to work at full capacity to meet the target. • To receive the bonus, design has to work at full capacity too. • If it doesn’t work at full capacity, the production goal is not reached, and design incurs losses for every unit of effort expended. Every effort less than full capacity is dominated by not working. • Therefore design works at full capacity or not at all.
The teams choose • Manufacturing also faces the same choice. • If they both work at full capacity, profits equal the bonus less the cost of effort, which nets design $50,000. • Thus, it is better to work at full capacity than not at all, providing the other team does too. Management should stress this point. • We conclude the structure of incentive schemes (as well as the total amount) can have a big effect on behavior within groups.
Introduction to Nash equilibrium • The concept of equilibrium is that each player's behavior can be viewed as the outcome from him optimizing an individual objective function that is partly defined by the solutions of optimization problems solved by the other players. • Read Chapters 9 and 10 of Strategic Play.
Motivation for best replies • After eliminating those strategies which seem implausible because they are dominated, or iteratively dominated, we are sometimes still left with many possible strategic profiles. • In these cases we must impose more stringent assumptions on how players behave to reach a sharper prediction about the outcome of a game. • One approach is to argue that each player forms a conjecture about the other players’ strategies, and then maximizes his payoffs subject to this conjecture.
Defining a best reply • In order to develop this concept we first define the notion of a best reply, which means a pure or mixed strategy that maximizes the player's payoff given a strategic profile of choices made by the other players in the game. • One approach a player might consider to selecting a strategy is to determine which strategy maximizes her payoff given the strategies selected by the other players. • This is called her best reply.
Competition through integration • In this game, a specialized producer of components for a durable good has the option of integrating all the way down to forming dealership, but indirectly faces competition from a retailer, which markets similar final products, possibly including the supplier’s.
Supplier • The supplier’s profits are higher if the retailer only distributes. • The supplier makes higher profits by undertaking more downstream integration if the retailer integrates upstream, to avoid being squeezed. • If the retailer confines itself to distribution, then the best reply of the supplier is to focus on its core competency, and only produce component parts.
Retailer • The profits of the retailer on this item fall the more integrated is the supplier. • If the supplier assembles but does not distribute, then the retailer should also integrate upstream, and assemble too. • Otherwise the most profitable course of action of the retailer is to focus on distribution.
Nash equilibrium for integration game • The Nash equilibrium is Up (make components) and Left (distribute only). • To verify this claim note that if the retailer chooses to distribute only by moving Left, then the best response of the supplier is to only make components, by moving Up. • Similarly if the supplier only makes components, moves to Up, then the best response of the retailer is to specialize in distribution, moving Left. • By inspecting the other cells one can check there is no other Nash equilibrium in this game.
A chain of conjectures The Nash equilibrium is supported by a chain of conjectures each player might hold about the other. Suppose: S, the Supplier, thinks that R, the Retailer, plays Left =>S plays Up; R thinks that S thinks that R plays Left =>R thinks that S plays Up =>R plays L; S thinks that R thinks that S thinks that R plays Left =>S thinks that R thinks that S plays Up =>S thinks that R plays Left =>S plays Up; and so on.
Definition of Nash equilibrium • Consider an N player game, where sn is the strategy of the nth player where 16 n6 N. • Also write s-n for the strategy of every player apart from player n. That is: s-n = (s1, . . ., sn-1, sn+1, . . ., sN). • Now ask whether sn is the best response of n to s-n for all n? If so, nobody has an incentive to unilaterally deviate from their assigned strategy. • This is called a Nash equilibrium.
Reduced form of Product differentiation • The low quality producer’s strategy of “reduce price of existing model” is dominated by any proper mixture of the other two strategies. • One we eliminate that strategy we are left with a two by two matrix.
Best reply in product differentiation game – high quality producer • With regards the high quality producer in the product differentiation game: • The best reply to the low quality producer introducing a new product is to hold a sale. • The best reply to the low quality producer upgrading an existing model is to do the same thing.
Best reply in product differentiation game – low quality producer • With regards the low quality producer: • The best reply to the high quality producer introducing a new product is to upgrading the existing model. • The best reply to the high quality producer upgrading the existing model is to introduce a new model. • The best reply to the high quality producer holding a sale is to introduce a new model.
Five Rules Rule 1: Look ahead and reason back. Rule 2: If there is a dominant strategy, play it. Rule 3: Discard dominated strategies. Rule 4: Iteratively eliminate dominated strategies. Rule 5: If there is a unique Nash equilibrium, then play your own Nash equilibrium strategy.
