General idea. Firms can use competition between agents for. We mainly focus on incentives. 1 incentive and. 2 selection purposes 3 / 101

Transcription

1 3 Tournaments

2 3.1 Motivation

3 General idea Firms can use competition between agents for 1 incentive and 2 selection purposes We mainly focus on incentives 3 / 101

4 Main characteristics Agents fulll similar tasks performances are comparable The principal xes prizes before agents compete The best performing agent gets the highest prizes, the second best gets the second highest prize and so on 4 / 101

5 Examples how rms use contests Job-promotion tournaments Firms often award a promotion to the best performer in the lower-level job Potential conict between incentives and selection: e.g., in an R&D organization, the best researcher does not need to be the best manager of the research group Forced-distribution systems for performance appraisals E.g., specic percentages for each rating Helps to reduce the following problems: managers tend to give many employees the same rating; or some are lenient, while others are strict Sales contests Sharing of a xed bonus pool (e.g., in big Japanese companies) 5 / 101

6 UPS UPS has promoted from within for generations. This includes: Part-time workers moving into full-time positions Non-management employees moving into management positions Supervisors and managers moving into positions of greater responsibility Well over 58 percent of the current full-time drivers were once part-time employees More than 77 percent of the full-time managers (including most vice presidents) were once nonmanagement employees 6 / 101

7 Advantages of tournaments + performance signals do not need to be veriable + Only ordinal information needed + Filtering of common noise (risk costs may decrease) such as general working conditions or rm reputation + Principal perfectly knows wage costs ex-ante 7 / 101

8 Disadvantages of tournaments Contradicts the cooperative idea of teamwork Sabotage Horizontal collusion Cheating (breaking the rules) and inuence activities (e.g., bribing the supervisor) Incentives may be decreased by: External job oers, intermediate information, or heterogeneity 8 / 101

9 3.2 Basic model

10 Model Two risk-neutral and homogeneous agents A and B q i = e i + ε i : output (in monetary terms) of agent i = A, B e i : agent i's eort ε i : exogenous noise term ε A and ε B are independently and identically distributed (i.i.d.) with distribution function F (ε) and corresponding density function f (ε) = F (ε) q i is observable, but not veriable (and thus not contractible) There is moral hazard: Neither e i nor ε i are observable ei and ε i are therefore also not veriable and not contractible Remark: The model is due to Lazear and Rosen (1981) 10 / 101

11 Timing 1 The principal suggests a contract 2 Each agent can either accept or reject If an agent rejects, he receives the reservation utility ū If an agent accept accepts, he chooses an eort level 3 The noise terms realize, outputs are determined, and payments are made according to the contract Remarks: If not explicitly stated dierently, we suppose throughout the course that the principal always wants to make sure that the agents accept Agents' timings are parallel: i.e., if agents accept, they simultaneously choose eorts 11 / 101

12 Winning probabilities The probability that agent i wins (and not j, where i, j {A, B} and i j) is prob{q i > q j } = prob{e i + ε i > e j + ε j } = prob{ε j ε i < e i e j } = G(e i e j ) G( ): distribution function of the composed random variable ε j ε i, with corresponding density g( ) Examples: 1 ε i, ε j N ( µ, σ 2 ), then εj ε i N ( 0, 2σ 2 ) 2 ε i, ε j are uniformly distributed, then G( ) is a triangular distribution Remark: There are also models with multiplicative noise, i.e., q i = ε i e i 12 / 101

13 Exercise 3.1 Suppose ε i, ε j N (0, 1/2) and agents choose eorts e j = 1 and e i = 2 Determine the probability that agent i wins How does this probability change if ε i, ε j N (10, 1/2)? How does this probability change if ε i, ε j N (0, 2)? 13 / 101

14 Eort costs, prizes, liability c(e i ): agent i's eort costs (in monetary terms) Assumptions: c is twice continuously dierentiable, c(0) = c (0) = 0, c (e i ) > 0 for all e i > 0, lim ei c (e i ) =, c (e i ) > 0 w 1, w 2 : winner prize and loser prize, respectively In case two or more agents have the same output, their places will be determined by drawing lots If g( ) has no mass pointswhich we assume henceforththe probability that agents have the same output is zero There is unlimited liability: i.e., tournament prizes can take every value, e.g., can also be negative 14 / 101

15 First-best solution To have a benchmark, we solve for the rst-best, where eorts are contractible The Lagrangian is L (e, w) = E[q i ] w λ( w + c(e i ) + ū), where w is the xed payment to agent i Since E[q i ] = e i + E[ε i ], we can use the result from the introduction: e FB solves 1 = c ( e FB) Intuition: In the rst-best, an agent's eort is chosen such that the marginal product equals the marginal eort costs 15 / 101

16 Agents' eort choices Suppose next that eorts are not contractible, outputs are observable but not contractible, agents have accepted and thus compete for the prizes w 1 and w 2 Agents simultaneously choose their eort levels Thus, agent i takes agent j's eort as given when choosing e i Agent i's expected utility is EU i = (w 1 c(e i )) G(e i e j ) + (w 2 c(e i )) (1 G(e i e j )) = w 2 + w G(e i e j ) c(e i ), where w := w 1 w 2 denotes the prize spread Similarly, agent j's expected utility is EU j = w 2 + w (1 G(e i e j )) c(e j ) 16 / 101

17 Equilibrium Each agent maximizes his expected utility over his eort choice Assuming that a pure-strategy equilibrium exists, it is given by the rst-order conditions w g(e i e j ) = c (e i ) w g(e i e j ) = c (e j ) As both LHSs are identical, the unique and symmetric equilibrium is e i = e j = e with w g(0) = c (e) (IC) 17 / 101

18 Discussion of (IC) Equilibrium eort is increasing in The prize spread w The expression g(0) as an inverse measure of luck The atness of the cost function c( ) Since in equilibrium both agents invest the same amount of eort, the tournament winner is purely determined by luck: winning probability G(0) = / 101

19 ε i can be interpreted as individual productivity shocks (the noise is at the productivity level) or individual measurement error of the supervisor (the noise is at the measurement level) or individual talent that is unknown to everyone ex-ante (the noise is at level of the agents) In the last case: the agent with the highest (unknown) talent wins Perfect selection of agents, e.g., in a promotion tournament

20 Principal's optimal tournament contract Principal maximizes the expected net prot E [q A + q B ] w 1 w 2 subject to: the incentive constraint (IC) and the participation constraint w 1 + w 2 2 c(e) ū (PC) 20 / 101

21 Equivalence This is equivalent to the case where the principal maximizes the expected net prot per agent subject to: E [q i ] w 1 + w 2 2 the incentive constraint (IC) and the participation constraint w 1 + w 2 2 c(e) ū (PC) 21 / 101

22 Lagrangian L (e, w 1, w 2 ) = e + E[ε] w 1 + w 2 ( 2 λ 1 w ) 1 + w 2 + c(e) + ū 2 λ 2 ( (w 1 w 2 ) g(0) + c (e)) 22 / 101

23 Exercise 3.2 Solve the Lagrangian. Does the (PC) bind in equilibrium? What is the intuition? 23 / 101

24 Remarks The optimal prize spread w = c (e ) g(0) are high (i.e., g(0) is small) Exercise 3.3 What is the intuition? is large if random inuences On average, each agent is just compensated for his eort costs and his forgone outside option ū However, ex-post the winner is strictly better o than the loser This is why the tournament creates incentives 24 / 101

25 Comparison to rst-best The solution satises 1 = c (e ) Thus, e = e FB Also the principal's wage costs are equal to the rst-best level: w 1 + w 2 = 2c(e ) + 2ū The principal's expected utility thus equals the rst-best level as well That is, despite that eorts are not observable and outputs are observable, but not veriable, the optimal tournament contract leads to the rst-best solution in the basic tournament model (homogeneous, risk-neutral agents with unlimited liability) 25 / 101

26 Exercise 3.4 What is the intuition? 26 / 101

27 Credibility of contracts The principal can observe outputs q i, q j But outputs are not veriable (and thus not contractible) The principal could assign the loser prize w 2 to both agents But this in contradiction to the contract, which species that one agent receives the winner- and the other the loser-prize Agents can go to a court and force the principal to pay the prize w 1 to one of them Principal has to assign the winner prize to one agent 27 / 101

28 Alternatively, the principal could assign the winner-prize w 1 to the loser (the agent with the lower output level) and the loser-prize w 2 to the winner (the agent with the higher output level) Agents cannot go to a court, since outputs are not veriable The principal's total payment to the agents is the same when she assigns the winner-prize to the winner and the loser-prize to the loser Since the principal is indierent, we assume that she prefers to follow the latter strategy With repeated interaction, the principal benets from having a reputation to follow this strategy and this strategy is then strictly optimal

29 Bargaining power We assumed that the principal has all the bargaining power (as in the standard principal-agent model) Suppose now that the agents have all the bargaining power, i.e., can x w 1 and w 2 (plausible in a competitive labor market with zero expected prots in equilibrium) Claim: The result does not change, i.e., the rst-best eort is still implemented Hint: replace the (PC) with the nonnegative-prot condition E[q i ] w 1 + w (NPC) 29 / 101

30 Exercise 3.5 Prove the claim (exploit that in equilibrium both agents invest the same eort) 30 / 101

31 Does a pure-strategy equilibrium exist? The SOCs for the agents' eort choices are w g (e i e j ) c (e i ) < 0 and w g (e i e j ) c (e j ) < 0 for all e i, e j Thus, only if the cost function is suciently convex (meaning that the marginal eort costs increase suciently fast) and/or the density is suciently at (meaning that there is sucient noise in the tournament) are the SOCs satised 31 / 101

32 Note If a SOC or the SOCs is/are violated, then we cannot be sure that the agents optimally choose the eorts given by the FOC, i.e., the (IC) But: This does not mean that the agents choose eorts dierent from that given by the (IC) in any case Example: EU i bell-shaped in e i (EU i is not concave SOC not satised, but maximum is still given by the FOC) 32 / 101

33 Zero noise Suppose there is zero noise (i.e., ε A = ε B = constant) Exercise 3.6 Does a pure-strategy Nash equilibrium exist? Why or why not? Concentrate on the case where both agents invest the same, positive eort level 33 / 101

34 Mixed-strategy equilibrium In a mixed-strategy equilibrium each agent chooses an optimal probability distribution over the interval [0, ē] ē is implicitly given by c(ē) = w Let G j (e i ) be the probability that j exerts e i or less eort Then i's expected utility is EU i (e i ) = w 2 + w G j (e i ) c(e i ) Agent i must be indierent between choosing the lowest possible eort e i = 0, ē, and any eort between (otherwise, she would not mix over [0, ē]) 34 / 101

35 If i plays e i = 0 she loses for sure Thus EU i (0) = w 2 c(0) = w 2 Therefore, EU i = w 2, i.e., in the mixed strategy equilibrium each player's expected utility equals w 2 If i plays e i (0, ē) she may win Her utility is then EU i (e i ) = w 2 + w G j (e i ) c(e i ) By the arguments from above we must have EU i (0) = EU i (e i ) Solving yields G j (e i ) = c(e i) w

36 Common noise Let agent i's output, where i {A, B}, be q i = e i + ε i + η Then, i's winning probability is prob (e i + ε i + η > e j + ε j + η) = prob (ε j ε i < e i e j ) Hence, the common noise η is completely eliminated 36 / 101