Gender Dierences. Experimental studies suggest that. Gneezy, Nierderle, and Rustichini (2003) let subjects solve computerized mazes

Transcription

1 3.3 Some Empirics

2 Gender Dierences Experimental studies suggest that Men are more strongly motivated by competitive incentives than women or Men are more eective in competitive environments than women Gneezy, Nierderle, and Rustichini (2003) let subjects solve computerized mazes Men and women perform equally well under piece rates But men perform better than women under competitive incentives 38 / 101

3 Gneezy and Rustichini (2004) show that gender dierences are present at a young age: In a 40 meter dash, nine-year old boys run much faster in a race than when they run alone Girls run as fast as boys when running alone, but do not increase their speed under competition Remark: Gender dierences in competition is a possible explanation for the underrepresentation of women in top positions (in addition to gender dierences in ability and preferences and gender discrimination)

4 Another study Delfgaauw et al. (2013) conducted a eld experiment in a Dutch retail chain Background: Discount retail chain selling clothing, shoes, and sports apparel 128 geographically dispersed stores, 1547 employees Average store has 12 employees, of which 85% are female (no stores with a majority of male employees) Slightly less female-led stores than male-led stores Store employees earn an hourly wage Store managers earn about 45% more, performance pay based on sales 40 / 101

5 Experimental Set-Up The authors designed the following sales contest among stores (i.e., tournament between teams): Performance measure: percentage sales growth compared to the same period last year Winner prize: 75 Euros for the store manager and each employee ( 5% of an employees monthly wage) Second prize: 35 Euros The stores competed in a (relatively homogeneous) pools of ve during a period of six weeks in the years 2007 & 2008 They received weekly feedback on sales in all stores Store managers and employees did not know that they were participating in an experiment Store managers had to inform employees in their stores about the 41 / 101

6 Results Tournament has a signicant positive eect on weekly sales growth On average, eect does not dier between stores with a male manager and those with a female manager No evidence that the gender composition of teams matters However: in male-led (female-led) stores, the eect of competition increases in the share of male (female) employees 42 / 101

7 Two Possible Explanations 1 Eectiveness of communication: Managers succeeded in making the competition appeal to employees of their own sex, but failed to do so to employees of the opposite sex 2 Avoiding free-rider problems may be easier if a manager and a large part of the store are of the same gender 43 / 101

8 3.4 Horizontal Collusion

9 The idea Consider the basic model, where the SOCs hold Thus, we have a symmetric and pure Nash equilibrium, where both agents invest a positive level of eort Hence, each agent's winning probability is 1/2 And both agents have positive eort costs If both invest zero eort, then each agent's winning probability is still 1/2 And both agents have zero eort costs 45 / 101

10 Therefore, both agents can gain from horizontal collusion By collectively reducing eort, they can save eort costs Collusion is illegal Agents are thus unable to write binding contracts Collusion must therefore be self-enforcing to be stable Note: we concentrate on horizontal collusioni.e., collusion between agents Another form of collusion (which we do not consider) is vertical collusion: an agent and a principal collude at the expense of the remaining agent

11 Model Reconsider the basic model we know from Chapter 3.2 Suppose agents can observe each other's eort choices at the end of each period Simplifying assumptions: w2 = 0 Each agent can only choose between two eort levels: e { e L, e N} We have e L < e N ( e N, e N) is the Nash equilibrium in the static tournament game, see Chapter 3.2 Under which conditions can the collusive outcome ( e L, e L) be stable? 47 / 101

12 Static game Recall: G(e i e j ) is i's winning probability The tournament can be characterized by a simple game: e N e L e N w 1 2 c ( e N), w 1 G ( e) c ( e N), w 1 2 c ( e N) w 1 G ( e) c ( e L) e L w 1 G ( e) c ( e L), c ( e L), 2 w 1 G ( e) c ( e N) w 1 c ( e L) 2 w 1 with e := e N e L > 0 so that G ( e) < 1 2 < G ( e) 48 / 101

13 Gains Obviously, both workers would be better o by choosing ( e L, e L) instead of ( e N, e N) c ( e N) c ( e L) is the individual gain from colluding Since ( e N, e N) is a Nash equilibrium, the agent must be at least as well o playing e N instead of e L, given that the other agent plays e N We consider the basic model, where the SOCs hold Therefore, it is strictly optimal to play e N for an agent, given that the other agent plays e N 49 / 101

14 Formally, we must have w 1 2 c ( e N) > w 1 G ( e) c ( e L) We can rewrite this as ( ) 1 2 G ( e) w 1 > c ( e N) c ( e L) (1) }{{} =: G G also represents the increase in the winning probability when deviating from collusion

15 By symmetry we have that 1 G(e i e j ) = G ( (e i e j )) We can thus rewrite G := 1 2 G ( e) = 1 2 (1 G ( e)) = G ( e) 1 2 > 0

16 Playing e N is strictly optimal also when the other agent plays e L : w 1 G > c ( e N) c ( e L) ( w 1 G ( e) 1 ) > c ( e N) c ( e L) 2 w 1 G ( e) c ( e N) > w 1 2 c ( e L) Note that the rst line is (1); the second line is obtained using G = G ( e) 1, see above; the third line is obtained by 2 rewriting the second line

17 Dominant strategy To summarize, playing e N is always the best response Hence, playing e N is a strictly dominant strategy This is a kind of a prisoner's dilemma Collusion is thus never stable in the static game 53 / 101

18 Dynamic game: nite number of repetitions Suppose the game is repeated T times, with T > 1 and nite In period T, both agents play ( e N, e N), no matter what they played before Given this, agents also play ( e N, e N) in period T 1... By backward induction, the agents always play ( e N, e N) Thus, collusion is also not stable with a nite number of repetitions 54 / 101

19 Dynamic game: innite horizon Agents play innitely often Alternative: agents do not know the nal period of the game Backward-induction argument does not apply Let δ (0, 1) denote the common discount factor Possible interpretations of δ: 1 1 Measure of agents' time preferences: δ := ; the smaller 1+interest rate is δ, the stronger are the preference for present payos 2 Continuation probability: Probability of meeting the other agent also in the next period in a tournament 55 / 101

20 Strategies We assume that each player chooses between the two strategies NF and GT : NF : choose Nash e N forever This can be interpreted as the unfriendly strategy GT : start with e L ; choose e L as long as the other player has chosen e L in the period before; if the other player chooses e N, then also switch to e N and stick to e N forever This can be interpreted as the friendly strategy The strategy is known as grim-trigger strategy 56 / 101

21 Combinations There are four combinations: (NF, NF ), (NF, GT ), (GT, NF ), (GT, GT ) Which combinations can be sustained as equilibria in the innite-horizon game? Simplify notation: α := w1 G ( e) c ( e N) β := w 1 2 c ( e L) γ := w 1 2 c ( e N) φ := w1 G ( e) c ( e L) Note that we have α > β > γ > φ 57 / 101

22 Present values Each player wants to maximize the present value of his income (=wage - eort costs) Consider, for example, the combination (NF, NF ) where each player receives γ in any period Exercise 3.7 What is the associated present value? Calculate also the present values for the other combinations 58 / 101

23 Summary of present values NF GT NF GT γ 1 δ, γ 1 δ φ + δ γ 1 δ, α + δ γ 1 δ α + δ γ 1 δ, φ + δ γ 1 δ β 1 δ, β 1 δ 59 / 101

24 Equilibria The outcome of the innite-horizon game can be characterized as follows: (NF, NF ) is always an equilibrium (reason: no single player wants to deviate) (NF, GT ) and (GT, NF ) can never be an equilibrium (reason: playing GT as response to NF is never optimal) (GT, GT ) is an equilibrium if and only if δ w 1G( e) c ( e N) w c ( e L) w 1 G( e) w 1 2 = w 1 G [ c ( e N) c ( e L)] w 1 G (2) 60 / 101

25 Exercise 3.8 Prove (2). Hint: Denoting the present value of agent i by V i, explore (GT,GT ) (NF,GT ) when V i V i holds 61 / 101

26 Interpretation Agent j plays strategy GT Is it better for agent i to play NF or GT? In the short-run, strategy NF is better for agent i than strategy GT : If agent i plays strategy NF, then he yields the maximal possible expected utility in period t = 1, namely α := w 1 G ( e) c ( e N) In contrast, if agent i plays strategy GT, then he yields only an expected utility of β := w 1 2 c ( e L) in period t = 1 62 / 101

27 However, in the long run, strategy NF is worse for agent i than strategy GT : If agent i plays strategy NF, then he yields an expected utility of γ := w 1 2 c ( e N) in all periods t 2 In contrast, if agent i plays strategy GT, then he yields an expected utility of β := w 1 2 c ( e L) in all periods t 2 If δ is suciently large, then the long run is suciently important for agent i such that he prefers strategy GT

28 Multiple equilibria If condition (2) holds, there are two equilibria: (NF, NF ) and (GT, GT ) It is an equilibrium that agents never collude and play Nash And it is also an equilibrium that agents always collude The latter equilibrium is Pareto superior Nontechnical intuition: Each agent's winning probability in a period is 1/2 in both equilibria An agent has lower eort costs in case of collusion since e L < e N Thus, under condition (2), we expect that the agents collude 64 / 101

29 Exercise 3.9 How can the principal prevent collusion? 65 / 101

30 Excursus: Collusion of rms Suppose that there are two identical rms with constant marginal production costs They compete either in prices or in output In the competitive outcome, their prots are lower than the monopoly prot Firms have an incentive to collude But collusion is stable only when δ is suciently high, i.e., if rms are likely to stay in the market and use a high discount factor The insights are thus the same as with agents in a tournament 66 / 101

31 3.5 Sabotage

32 The idea Important disadvantage of tournaments (compared to individual incentive schemes): a player can increase his expected payment by lowering the performance of others Sabotage = any activity that is intended to reduce the performance of the rival in the tournament Examples: Withholding information, uncooperative behavior, bullying / 101

33 Model New output functions: q i = e i+ + ε i e j q j = e j+ + ε j e i e i+ denotes agent i's eort choice e i is agent i's sabotage activity, which is directed against the other agent c + (e i+ ): eort cost function, with c + > 0 and c + > 0 c (e i ): sabotage cost function, with c > 0 and c > 0 69 / 101

34 Objectives Probability that i wins is prob(q i > q j ) =prob (e i+ + ε i e j > e j+ + ε j e i ) =prob (e i+ + ε i + e i > e j+ + ε j + e j ) =prob (e i+ + e i e j+ e j > ε j ε i ) =G(e i+ + e i e j+ e j ) The agents' objective functions are thus EU i =w 2 + G(e i+ + e i e j+ e j ) w c + (e i+ ) c (e i ) EU j =w 2 + [1 G(e i+ + e i e j+ e j )] w c + (e j+ ) c (e j ) 70 / 101

35 First-order conditions The rst-order conditions are g(e i+ + e i e j+ e j ) w = c +(e i+ ) g(e i+ + e i e j+ e j ) w = c (e i ) g(e i+ + e i e j+ e j ) w = c +(e j+ ) g(e i+ + e i e j+ e j ) w = c (e j ) 71 / 101

36 Equilibrium Suppose that a pure-strategy equilibrium exists Then the equilibrium must be symmetric: e i+ = e j+ =: e + and e i = e j =: e The equilibrium values solve g(0) w = c +(e + ) g(0) w = c (e ) Observe that both productive eort and sabotage increase in w and g(0) 72 / 101

37 Empirical Evidence from Soccer Garicano and Palacios-Huerta (2006) analyze an incentive change in soccer leagues (tournament between teams) Historically: soccer teams engaging in league competition have been rewarded with 2 points for winning a match and 1 point for tying Since 1994: 3 points for winning and 1 point for tying w Objective of the FIFA: encourage attacking, high-scoring matches 73 / 101

38 Findings Measures of desired, attacking eort (shot attempts on goals, corner kicks) increased (by 10 percent) Measures of sabotage activities (fouls, unsporting behavior punished with yellow cards, more defenders) also increased (by 12.5 percent) Net result: Number of goals did not change 74 / 101

39 3.6 Heterogeneous Agents

40 The idea Agents usually dier in their abilities, skills, experience etc. To model this, let ε i, ε j N (µ, σ 2 ) Then ε j ε i N (0, 2σ 2 ) Output is q i = e i + a i + ε i a i : agent i's exogenous ability (known to the agents) 76 / 101

41 Objectives Probability that i wins is prob(q i > q j ) =prob (e i + a i + ε i > e j + a j + ε j ) =prob (ε j ε i < e i + a e j ) =G (e i + a e j ) a := a i a j is the ability dierence Agents objective functions are EU i = w 2 + wg (e i + a e j ) c (e i ) EU j = w 2 + w (1 G (e i + a e j )) c (e j ) 77 / 101

42 First-order conditions The rst-order conditions are wg (e i + a e j ) = c (e i ) wg (e i + a e j ) = c (e j ) Again, we have a symmetric equilibrium (if it exists) So e i = e j =: e But now e depends on a c (e) = wg ( a) (IC) 78 / 101

43 Results The density of the normal distribution is bell-shaped and has a maximum at its mean Here: mean is zero Therefore, the higher the degree of heterogeneity between the workers (i.e. the larger a ), the lower will be the agents' equilibrium eort levels Exercise 3.10 What is the intuition? 79 / 101

44 Remark The same technical result would apply if agents are homogenous, there is intermediate information, and one of the agents had a lead a Idea: The agents' eort choices depend on the lead Thus, the principal does not want to reveal intermediate information (This holds if g has a maximum at zero; which is true for many sensible density functions) 80 / 101

45 Optimal tournament design Consider again the model with heterogenous agents Without loss of generality, let agent i be the favorite: a > 0 If the principal can observe the agents' abilities she will impose the handicap a on favorite i That is, i wins only if q i q j is larger than the handicap a Then the agents' eort choice solves c (e) = wg (0) and no longer c (e) = wg ( a) Since g (0) > g ( a), e increases 81 / 101

46 Unknown abilities Suppose the principal does not know the agents' abilities Both agents choose the same eort levels But favorite i's winning probability is G ( a) > 1 2 Therefore while for agent j the participation constraint is binding, it does not bind for i That is, EU i > ū Agent i earns a positive rent in the optimum Solving the principal's problem yields that she optimally implements less than rst-best eort: e < e FB This highlights the problems of tournaments in case of unobservable heterogeneity 82 / 101

47 3.7 Risk Aversion

48 Costless incentives In the basic model and all variants we explored (except in the last, with unknown abilities) setting incentives is costless Technically, the Lagrange parameter of the incentive constraint is zero Intuition: Setting incentives necessarily involves risk (an agent may win or lose with some probability) Since the agent is risk neutral, his marginal utility of money is constant Thus, the principal does not have to compensate the agent for risk The participation constraint just requires that the expected prize or wage is at least as great as the agent's outside option plus his eort costs 84 / 101

49 Costly incentives As we will see next, setting incentives is costly if the agent is risk averse In order to participate, each agent demands a risk premium 85 / 101

50 Introducing risk aversion Let again q i = e i + ε i Each agent has the same additive-separable utility function Let v ( ) be quadratic: u (w i, e i ) = v (w i ) c (e i ) v (w i ) = w i rw 2 i r > 0: indicating a player's degree of risk aversion wi < 1/(2r): guarantees a non-decreasing utility function 86 / 101

51 Objectives EU i (w i, e i ) = E [w i ] re [ ] wi 2 c (ei ) = w 2 + wg (e i e j ) r ( w 2 (1 G (e 2 i e j )) + w 2 G (e 1 i e j ) ) c (e i ) EU j (w j, e j ) = E [w j ] re [ ] wj 2 c (ej ) = w 2 + w (1 G (e i e j )) r ( w 2 G (e 2 i e j ) + w 2 (1 G (e 1 i e j )) ) c (e j ) 87 / 101

52 Equilibrium An equilibrium in pure strategies is characterized by the FOCs w (1 r (w 1 + w 2 )) g (e i e j ) = c (e i ) w (1 r (w 1 + w 2 )) g (e i e j ) = c (e j ) Equilibrium eorts decrease in the risk aversion parameter r The risk premium is also increasing in r 88 / 101

53 Cost minimization Cost minimization is necessary for prot maximization We can thus decompose the principal's problem into two steps: 1 Choose prizes to minimize costs, given that certain eorts should be implemented 2 Choose the implemented eorts to maximize prots One sometimes only solves the cost-minimization problem, since this already provides information about important properties of the optimal contract This is often the case in Chapter 4 89 / 101

54 Exercise 3.11 Look at the cost-minimization problem of the principal Suppose that the SOCs are satised, c(e) = e 2 /2, g(0) = 1/2, r =.1, and ū = 0 What prizes does the principal optimally choose to minimize her costs and implement e 1 = e 2 = 1? Hint: You can use wolframalpha or some other website or program to solve the quadratic equations 90 / 101

55 3.8 Further Applications

56 Examples Tournament theory has many applications outside rms. Examples are: Rent-seeking: Spending wealth on political lobbying to increase one's share of existing wealth without creating wealth (practical example: manipulation of regulatory agencies to gain monopolistic advantages) Sports contests War between countries Legal conicts (lawsuits) Election campaigns Procurement contests (a rm or a government needs a certain product or service and potential suppliers compete for the order) 92 / 101

57 Tullock contests The economist Gordon Tullock, cf. Tullock (1967, 1980), developed a certain class of contests/tournaments. Consider the most basic variant Two agents Eort costs are c(e i ) = e i w 1 > 1 and w 2 = 0 The probability that agent i wins is e i e i +e j Alternative interpretation: i receives the share e i e i +e j for sure 93 / 101

58 Solution Agent i maximizes The FOC is The SOC is d 2 EU i de 2 i EU i = ( deu i 1 = de i e i + e j e i e i + e j w 1 e i e i (e i + e j ) 2 ) w 1 1 = 0 = (e i + e j ) 2 w 1 (e i + e j ) 2 w 1 + 2e i (e i + e j ) 3 w 1 sign = (e i + e j ) 2 + e i (e i + e j ) 3 sign = (e i + e j ) + e i = e j 0 94 / 101

59 In the symmetric equilibrium e i = e j =: e The FOC is then ( 1 2e e (2e) 2 ) w 1 1 = 0 Solving yields Thus, in equilibrium 2c(e) = w 1 e = w 1 4 Interpretation (in case of rent-seeking, where eorts are unproductive): half of the prize is wasted 2

60 Foundation of Tullock contests q i = ε i e i ε i is exponentially distributed Density of ε i is thus: f (ε i ) = { λ exp{ λεi } for ε i 0, 0 otherwise. We prove in class that i's winning probability is then e i e i +e j 96 / 101

61 Application Natural resource wealth is a `curse' rather than a benet to society when property rights are not dened or respected and the wealth becomes a rentseeking prize (Congleton, Hillman, and Konrad 2008) In the basic model, see above, welfare (=prize minus eort costs) is increasing in the prize In a more complicated model, this need not be the case. Reasons: Agents forego productive, non-rent-seeking activities Rent-seeking may deter innovative activities 97 / 101

62 Anecdotal evidence from Nigeria Nigeria is rich in oil (11th largest proven oil reserves in the world) Income became highly concentrated during the oil price run-up between 1970 and the early 2000s By 2000, the share of income controlled by the richest 2% of the population equaled that of the poorest 55% In 1970, the richest 2% earned as much as the poorest 17% The fraction of Nigerians who subsist on $1 per day or less rose from 26% to 70% over the same period 98 / 101

63 Norway et al. By contrast, institutions that have been relatively eective in discouraging rent-seeking activity can explain the more favorable outcomes in resource-rich countries such as Norway, Chile, Malaysia, and Botswana Remark: With rent-seeking, we do not want that agents invest eort. This is in contrast to the case where eorts are productive, e.g., where rms want to incentivize employees 99 / 101

64 3.9 More than two Agents

65 Idea Could we allow for more than two agents? This is straightforward in case of a Tullock contest With n 2 agents, let the probability that agent i wins be Note that we can also extend the other tournament models e i n k=1 e k 101 / 101