Risk-Bearing in a Winner-Take-All Contest

Transcription

1 Gakushuin Economic Paers, Vol.45, No., ril 008, Gakushuin University, Tokyo, Jaan Risk-Bearing in a Winner-Take-ll Contest Dimitry Rtischev bstract arsimonious model is used to exlore the risk-bearing decision under a ayoff structure that emhasizes relative erformance. Equilibrium betting amounts are derived for layers who start with unequal endowments and face a lottery that offers either a ositive or negative exected return. f the lottery offers negative exected return, disadvantaged layers are willing to risk a ortion of their endowment, and this induces advantaged layers to also gamble, defensively. lthough there are equilibria in which the advantaged reemtively gamble more than the disadvantaged, in the robust equilibrium it is the disadvantaged who make the larger bets. f the lottery offers ositive exected return, there are equilibria in which the advantaged invest less than the disadvantaged, but full investment by all layers is a more robust equilibrium.. ntroduction The logic of the risk-bearing decision under a ayoff structure that emhasizes relative erformance combines stochastic and strategic considerations in a comlex way. The comlication was lucidly illustrated by ron and Lazear s (990) discussion of a sailing race as a metahor for cometition among firms develoing new roducts. n a sailing race, because all nearby boats are affected by the same wind, the leader maximizes robability of winning by imitating the follower s course, even if the leader believes another course has shorter exected time to the finish. Conversely, the follower increases its winning robability by differentiating its course from the leader s, even if it believes the different course has longer exected finish time than the course the leader is on. lying this insight from sailing, ron and Lazear study equilibria in which a firm that is behind in given roduct market leaves it develo a substitute roduct, and thereby induces the current market leader to also switch to ursuing the risky substitute. Other contests in which strategic risk-bearing has been studied include research ortfolio selection by scientists and firms racing to be first (Dasguta and Maskin, 987), managers cometing for relative erformance comensation (Hvide, 00), gambling by individuals struggling for socioeconomic status (Gregory, 980; Robson, 99; Rosen, 997), and contests embedded in evolutionary settings (Robson, 996; Dekel and Scotchmer, 999). The common thread among these different contexts is that the logic of the Faculty of Economics, Gakushuin University, Tokyo, Jaan. Comments on an early version of this work by Ryuichiro shikawa, Mitsunobu Miyake and Mamoru Kaneko are gratefully acknowledged.

2 risk-bearing decision under a relative ayoff structure may drastically deart from standard risk/return considerations that revail when only absolute erformance matters. n this aer, we abstract away from alied contexts to exlore the essence of this henomenon in a arsimonious setting. We begin in the next section by describing a model in which two unequally-endowed risk-neutral agents choose a ortion of their endowments to risk in a lottery, knowing that to-rank in ost-lottery wealth distribution will be allocated a rize. We derive best resonses (Section 3) and equilibria (Section 4) for two tyes of lotteries: ositive exected return lotteries (reresenting investing) and negative exected return lotteries (reresenting gambling). n Section 5 we generalize the findings to an arbitrary number of layers and rizes. Like ron and Lazear (990), we find that the worse-endowed agents roactively engage in risk-taking, and this induces the better-endowed agents to also take on risk. Unlike ron and Lazear, we find this without assuming any correlation in lottery outcomes across agents. We also extend the inquiry to consider the extent of risk-bearing under various timing structures and distinguish between ositive and negative exected return cases.. The model Two layers indexed by have initial monetary endowments such that. We will denote the endowment disarity by and refer to layer as rich and layer as oor. Each layer may bet any ortion of his endowment on a single indeendent draw of a binary lottery. fter both layers have laced their bets, two indeendent draws of the lottery are erformed. f a layer wins the lottery, he gets back his bet lus where ; if he loses he gets back nothing. Probability of a layer winning the lottery is.,, and are exogenously fixed arameters and are common knowledge. The lottery mas layers endowment vector and action vector to an achieved wealth random variable vector, where. The exected value of achieved wealth is, where is the exected rate of return offered by the lottery. We will consider both negative and ositive exected returns, referring to the former as gambling and the latter as investing. We will assume that, which ensures that it is ossible for the oor layer to leafrog the rich in the achieved wealth distribution. The utility function mas achieved wealth to utility. The baseline case is absolute wealth maximization, reresented by utility function () t is obvious that under this definition of utility, utility-maximizing strategies are : gainst this baseline, we will comare a winner-take-all contest in which ayoffs are given by the We do not analyze the secial case of.

3 Risk-Bearing in a Winner-Take-ll Contest Rtischev utility function () We assume that is a rize whose utility value exceeds that of any achievable wealth. Under both secifications of utility, higher achieved wealth corresonds to higher utility. However, under () achieved wealth can also imact utility through its role in allocating the rize to the relatively rich layer. 3. Best-resonse analysis 3. Contest success robabilities Following the conventional aroach to analyzing contests, we first characterize contest success robability as a function of layers actions. The two layers lottery draws can result in four distinct events:, where each event is a two-letter combination with layer listed first, and and reresenting each layer s win or loss in the lottery. Regardless of the amounts bet, the outcome necessarily results in. Which layer achieves more wealth in the other three cases deends on the bets as well as arameters and. Figure deicts one ossible geometry relating lottery outcomes to relative wealth. Figure and Table show the four ossible distinct geometries, enumerated as Cases through, and the contest success robability corresonding to each case. Since there are only 4 distinct contest success robabilities, the contest success function (CSF) in our model is a discrete-valued many-to-one function: Figure. The geometry of the contest with =5, =3, w=, z=, =0.5. The four ossible outcomes are the vertices of the rectangle. Outcome LW lies above the 45º line, which imlies m > m and layer obtains the rize. Outcomes LL, WW, and WL lie below the 45º line, which imlies m > m and layer obtains the rize. m 45º +w - LW LL WW WL -z +wz m

4 Figure. The four ossible geometries relating lottery outcomes and relative wealth. LW WW LW WW LL WL LL WL Case Case LW WW LW WW LL WL LL WL Case Case Table. The four ossible cases of how lottery outcomes relate to relative wealth. Case Lottery outcomes resulting in m m Lottery outcomes resulting in m > m WL, LL, LW, WW 0 WL, LL, WW -(-) LW (-) WL, LL - LW, WW WL, WW LL, LW - Figure 3 illustrates how the CSF artitions the domain into the four regions corresonding to four cases in Table. Our CSF differs markedly from those in conventional contest models (cf. Skaerdas 996, Hirshleifer 989). n a conventional contest model, layers comete for a rize by choosing effort level whose cost is given by an increasing function. The effort levels of all the layers jointly determine how the robability of winning the rize is distributed among the layers. CSF is continuous and satisfies (increasing own effort raises own robability of winning) and (increase in effort by a layer reduces the other layer s robability of winning). n contrast, our CSF is discrete-valued and decreasing at some oints in its domain. The ossibility that may fall as layer raises or may rise as another layer increases is a consequence of the downward exosure

5 Risk-Bearing in a Winner-Take-ll Contest Rtischev Figure 3. The CSF artitions its domain into four regions, which it mas to distinct success robabilities, corresonding to the four cases in Table. Some best resonses and equilibria deend on the relative sizes of and º, where º /w. /w - º -(-) z º - -(-) /w z < º º inherent to risking money in the lottery. For instance, if layer increases by a large enough amount, he may risk losing so much that the situation shifts from Case to Case, reducing his success robability from to. 3. Maximization of exected utility To maximize exected utility (), layer solves: Player s maximization roblem is analogous. Since, for large enough, the second summand dominates. f, maximum exected utility for layer occurs at the smallest that yields the highest. Thus, assuming the rize carries sufficient utility value, the roblem of maximizing exected utility facing each layer boils down to the following: bet the least amount that maximizes the robability of achieving more wealth than the other layer. Formally, for layer : f the lottery offers ositive exected return, a layer maximizes exected utility by risking the largest amount that maximizes the robability of achieving more wealth than the other layer. Formally, Regardless of whether exected returns are ositive or negative, the references of layer over the four ossible outcomes in listed in Table and deicted in Figure are: Geometrically, in Figure layer maximizes exected utility by finding the narrowest rectangle that includes the largest ossible number of vertices below the 45º line whereas layer seeks the shortest rectangle that includes the largest ossible number of vertices above the 45º line.

6 Since the cometition for success robability is a zero-sum game, layer holds the oosite references. 3.3 Best resonses 3.3. Negative exected return case The rich layer s best resonse deends on whether the oor layer gambles more or less than. f, then even if the oor wins the lottery he will not leafrog the rich in the achieved wealth distribution. The rich is not threatened and his best resonse is to abstain. However, if the oor gambles and the rich abstains, then with robability the oor will leafrog the rich. lthough the rich cannot comletely defend against this ossibility, he can reduce its robability from down to by gambling enough to establish Case. The smallest such defensive gamble is. Thus, rich layer s best resonse is (see Figure 4a): () We next turn to oor layer s best resonse when. f the rich abstains, the oor is best-off gambling, where is an arbitrarily small ositive constant. f, then the oor is best-off abstaining, hoing that the rich will lose his endowment advantage in his own gamble. f the rich gambles, then the oor is best-off matching and betting an additional to obtain Case. However, if the oor cannot afford this bet, then he is best-off aiming for Case by gambling as if the endowment ga has been reduced by the otential loss of the rich. Secifically, the oor layer s best resonse function is (see Figure 4b): where. f, the oor layer s best resonse is given by (see Figure 4c)

7 Risk-Bearing in a Winner-Take-ll Contest Rtischev Figure 4a. Rich layer s best-resonse ( <0) /w - º -(-) z < º Figure 4b. Poor layer s best-resonse ( <0, </) /w - º -(-) z º - -(-) /w z < º º Figure 4c. Poor layer s best-resonse ( <0, >/) /w - º -(-) z º - -(-) /w z < º º 3.3. Positive exected return case lthough he is risk-neutral and is facing a ositive-exected-return investment oortunity, the ossibility of losing his investment and thereby losing relative osition in the wealth distribution limits how much the rich layer invests. Secifically, if the oor invests less than, then the rich can guaran-

8 tee keeing to wealth rank (Case ) by investing no more that. f the oor invests more than, then the rich cannot obtain Case and must settle for the next-best Case. The rich layer s best-resonse function is (see Figure 5a) The oor layer may also under-invest in a ositive-exected-return oortunity in order to maximize robability of achieving to wealth rank. Secifically, if the rich invests an amount large enough that losing it would change relative wealth standing, the oor refers to limit his investment. But if the rich invests an amount so small that losing it will not change relative wealth standing, then the oor layer s Figure 5a. Rich layer s best-resonse ( >0) /w - º -(-) z < º Figure 5b. Poor layer s best-resonse ( >0, </) /w - º -(-) z º - -(-) /w z < º º Figure 5c. Poor layer s best-resonse ( >0, >/) /w - º -(-) z º - -(-) /w z < º º

9 Risk-Bearing in a Winner-Take-ll Contest Rtischev only hoe of leafrogging the rich is through investing himself, and then he might as well invest his entire endowment. Secifically, the oor layer s best resonse function is (see Figure 5b-c) 4. Equilibria The reaction curves derived in the revious section do not intersect. Therefore the game in which layers lace bets simultaneously has no Nash equilibrium in ure strategies. n this section, we examine equilibria of games in which layers lace bets sequentially. 4. Exogenous order of betting n the oor-first game, layer chooses, layer observes it and then chooses, and finally the two lottery draws are made, winnings (if any) aid out, and the rize awarded. n the rich-first game, the order of betting is reversed. Each game has a unique subgame erfect equilibrium. s shown in Figure 6a, when exected return is negative, the oor-first game has one equilibrium and the rich-first game has one of three ossible equilibria, deending on exogenous arameter values. nalogously, Figure 6b shows the four equilibria in the case of ositive exected return. n every equilibrium, there is a ositive robability that the oor leafrogs the rich and thereby obtains the rize. Equilibria in the oor-first game make intuitive sense. When exected return is negative, the oor layer gambles a substantial ortion of his endowment but the rich only a nominal amount. When exected return is ositive, both layers invest their entire endowments. Equilibria in the rich-first game, however, are counter-intuitive under some arameter values. Secifically, the rich may gamble more than the oor on a negative exected return lottery and invest less than the oor in a ositive exected return lottery. We next show that endogenizing the betting order eliminates the rich-first equilibria and retains only the oor-first equilibria. 4.. Endogenous order of betting Endogenizing betting order is roblematic for two reasons. First, comaring equilibrium outcomes in oor-first and rich-first games reveals that each layer refers to be the follower. This is because, given any arameter values, in the equilibrium of the game in which he is the follower, as comared to the equilibrium of the game in which he is the leader, a layer either obtains a higher success robability or the same success robability but with higher exected absolute wealth (due to less gambling when or more investing when. For this reason we cannot follow Baik and Shogren s (99) aroach to endogenizing the sequence of effort choices in a contest by letting each layer re-commit to a time at However, when and, layer is indifferent between being the leader or the follower.

10 Figure 6a. Summary of equilibria in the negative returns case ( <0). Each box reresents an equilibrium and corresonds to one of cases through in Table, as indicated on the to line within the box. The bottom two lines within each box indicate equibrium bets z and Poor-first Rich-first <º º /w + Endogenous-escalation </ 0 /w + >/ º 0 Rich gambles more than oor º ( -º)/w + Rich may gamble more than oor Figure 6b. Summary of equilibria in the ositive returns case ( >0). Poor-first Rich-first <º º </ Endogenous-escalation Rich may invest less than oor >/ - Rich may invest less than oor

11 Risk-Bearing in a Winner-Take-ll Contest Rtischev which he will announce his bet. The second difficutly is that if layers are allowed to resond to each others bets an unlimited number of times they will cycle endlessly raising and lowering their bets. straightforward way endogenize betting order in site of these difficulties is the following. Players take turns observing the other s latest bet and adjusting one s own bet. layer may increase his bet or kee it unchanged; decreasing a bet is disallowed. Thus, on the r th round of bet-setting, layer observes, the bet announced by layer on the revious round, and announces his bet. nitially all bets are zero:. The first time a round is comleted in which neither layer increases his bet, i.e. and, the bets are considered final and the game roceeds to the lottery drawings. First consider the case of negative exected return. On the first round, the rich layer does not bet because he is satisfied with the status quo and has no reason to bet reemtively since another round of betting is guaranteed if the oor layer bets something. The oor layer follows his best-resonse and bets. On the second round, the rich layer follows his best resonse by betting, thereby establishing Case. Player then considers following his best resonse function to raise his bet to and thereby establish Case. However, he knows that layer would then resond according by betting and thereby re-establishing Case. Such escalation could continue until and, at which oint the oor layer would reach his budget constraint and the rich layer would succeed in re-establishing Case. Thus, by escalating beyond, the oor layer gains no increase in success robability but shoulders more negative-return risk. By backward induction, the oor layer anticiates the inevitability of Case and does not escalate. By virtue of having a larger endowment, the rich layer has the last move in the endogenous-move game and the game has the same equilibrium as the oor-first game. n the case of ositive exected return, the rich layer starts out betting to establish his most-referred Case. The oor layer resonds by betting to establish Case or Case. The layers continue to escalate until both have fully invested their endowments and Case obtains. This equilibrium is identical to the equilibrium in the oor-first game. 5. Negative exected return case with many layers and rizes t is interesting to examine the case of negative exected return with layers. We assume that no two layers have the same endowment and index the layers such that. Let the number of rizes be, an integer such that. ll rizes are identical. Each layer who gets a rize obtains additional utility. The rich layers are layers ; each would obtain a rize if the lottery were not available and rizes were allocated based on endowed wealth rank. The oor are layers. ssume, which imlies that the oorest layer can afford a bet that gives him a chance of surassing the wealth of the richest. is sufficiently large in the sense of Section 3., which allows us to reduce each layer s exected utility maximization roblem to the roblem of betting the least amount to obtain the largest contest success robability. Bets Searate from the absolute rich/oor distinction, we will refer to a layer as richer (or oorer ) than layer if (or ).

12 are laced according to the same routine as described in Section 4.. Secifically, each layer has his turn to maintain or increase his bet according to some articular order such as. Bet-setting rounds are reeated until a full round is made without a single layer increasing his bet. Then all bets are considered final and an indeendent random draw of the lottery is held for each layer. fter the lottery, rizes are allocated according to rank in the achieved wealth distribution, one rize er layer until all rizes are awarded. Secifically, allocation begins with the layer(s) who have the most ost-lottery wealth and roceeds down one rank at a time. f, due to ties, the lowest rank for which there is at least one rize still available includes more layers than rizes remaining, the remaining rizes are allocated among these layers according to their rank in the endowment distribution. The simle case of and illustrates the logic of the many-layer many-rize contest. One strategy the single oor layer 3 may consider is to aim to surass the oorest rich layer by betting. The threatened layer will then resond with, giving the oor layer a success robability of (Case ). Player will not be threatened and will not gamble. However, this minimal gambling is not the best strategy for the oor layer. He can obtain more success robability by aiming to surass the richest layer by betting. Then the rich layers will resond with and. The oor layer stands to obtain a rize if he wins the lottery and either one or both of the rich layers lose in the lottery. This gives the oor layer a success robability of. None of the layers wants to escalate betting further since each can deduce that the escalation will be recirocated, otentially all the way u to the budget constraint, while the distribution of success robability would not change. This logic generalizes to any number of layers when there is only one oor layer, i.e.,. The oor layer will gamble, threatening the richest layer and thereby inducing every rich layer to gamble defensively, each seeking to avoid being the one who gives u his rize to the oor layer in case the oor layer wins the lottery. By inducing such defensive gambling among the rich, the oor layer gets a rize if he wins the lottery and at least one of the rich layers loses in the lottery, which imlies his success robability is the highest achievable:. dding an arbitrary number of oor layers to the revious case brings us to the general case of. By analogous logic it can be shown that in equilibrium each layer rich and oor bets. Each layer who wins the lottery joins the winner s circle with in wealth. Let be a random variable reresenting the number of lottery winners. There are four mutually exclusive, collectively exhaustive cases:. f, then each rich layer gets a rize and each oor layer goes rizeless.. f, then all winners get rizes and all losers go rizeless, regardless of who was rich and who was oor by endowment. 3. f, then all winners and the richest losers get rizes; all other losers go rizeless. 4. f, then the richest winners get rizes; the remaining winners and all losers go rizeless.

13 Risk-Bearing in a Winner-Take-ll Contest Rtischev For a oor layer, it is necessary but not sufficient to win the lottery to get a rize. This is because of case 4, wherein a oor layer may win the lottery but nevertheless go rizeless if or more richer layers also win the lottery. Secifically, for a oor layer, which can be exressed using the binomial distribution as follows: Remarkably, the robability of a oor layer getting a rize is indeendent of the total number of layers. ll that matters to a oor layer is the number of layers richer than him and the number of rizes; the number of layers oorer than him is irrelevant. n other words, how he ranks in terms of his endowment matters to a oor layer only by looking u the endowment hierarchy. For a rich layer, it is sufficient but not necessary to win the lottery to get a rize. This is because of case 3, wherein a rich layer may lose in the lottery but nevertheless get a rize if and fewer than losers are richer than. Secifically, for a rich layer, which can be exressed using binomial distributions as follows: lthough a oor layer s success robability is unaffected if additional layers oorer than him join the contest, a rich layer s success robability decreases with increasing number of oor layers. s Table and Figure 7 demonstrate, for a given number of rizes, having more oor layers corresonds to lower success robability for every rich layer. Each additional oor layer joining the contest does not affect other oor layers chances but redistributes some of the rich layers success robability to himself. n the limit as the number of oor layers grows large, the success robability of each rich layer aroaches. 6. Concluding remarks f the lottery were to be disabled in our model, the layer with the larger endowment would obtain the rize with robability one. Since in every equilibrium both layers have ositive robability of obtaining the rize, the betting is not mutually-offsetting and the lottery serves to narrow the oortunity ga between the layers regardless of whether it offers ositive or negative exected return. f the exected

14 Table. Contest success robabilities in a contest with endogenous order of bets, K = rizes and N = 3 to 0 layers. Probability of winning the negative exected return lottery is = 0.4. Keeing the number of rizes constant and adding a oor layer at the bottom of the endowment hierarchy redistributes contest success robabilities away from the two rich layers to the newly added oor layer, without affecting the success robability of the richer oor layers. Players N Prizes K Rich Poor Figure 7. Equilibrium success robabilities of the two rich layers in Table as a function of number of layers. (K=, N = 3,..., 0, = 0.4) return is negative, exected absolute wealth is sacrificed in all equilibria; if it is ositive, exected absolute wealth is enhanced, but, in some equilibria, not to the full extent ossible. f layers are solely concerned with maximizing absolute wealth er utility function (), the sign of the exected return constitutes necessary and sufficient information that each layer must know about the lottery to determine his otimal bet. f layers lace first riority on relative wealth and second riority on absolute wealth, as in utility function (), layers best resonse deends not only on but also on and on. n this case, the rate of return is neither necessary nor sufficient in-

15 Risk-Bearing in a Winner-Take-ll Contest Rtischev formation for selecting the otimal bet. nother way to see how the logic of the risk-bearing decision differs under absolute and relative wealth maximization is to consider layers references among lotteries. f offered a choice of two lotteries, a layer who maximizes absolute wealth er () would refer the lottery with the highest ositive exected rate of return or be indifferent if both lotteries offer negative exected return. Rate of return is necessary and sufficient information for an absolute wealth maximizer to decide which lottery he refers. For a relative wealth maximizer reresented by (), however, the rate of return is neither necessary nor sufficient to decide which lottery he refers be used in the contest. We have shown that with oor-first or endogenous sequence of bets, the equilibrium allocation of success robability is and. Since is the variance of the lottery and has a unique maximum at, the rich layer refers the lower-variance lottery with furthest from whereas the oor refers the higher-variance lottery with closest to. Unless both lotteries have the same variance, the two layers will refer different lotteries. This imlies that even if one lottery first-order stochastically dominates the other, one of the layers will nevertheless refer that it be chosen for use in the contest. References ron, Debra J. and Edward P. Lazear The ntroduction of New Products. merican Economic Review, 990, 80(), 4-46 Baik, Kyung Hwan and Jason F. Shogren Strategic Behavior in Contests: Comment. merican Economic Review, 99, 8(), Dasguta, Partha and Eric Maskin The Simle Economics of Research Portfolios. Economic Journal, 987, 97, Dekel, Eddie and Suzanne Scotchmer On the Evolution of ttitudes towards Risk in Winner-Take-ll Games. Journal of Economic Theory, 999, 87, 5-43 Gregory, Nathaniel Relative Wealth and Risk Taking: Short Note on the Friedman-Savage Utility Function. Journal of Political Economy, 980, 88:6, 6-30 Hirshleifer, Jack Conflict and Rent-Seeking Success Functions: Ratio vs. Difference Models of Relative Success. Public Choice, 989, 63, 0- Hvide, Hans K. Tournament Rewards and Risk Taking. Journal of Labor Economics, 00, 0:4, Robson, rthur J. Status, The Distribution of Wealth, Private and Social ttitudes to Risk. Econometrica, 99, 60:4, Robson, rthur J. The Evolution of ttitudes to Risk: Lottery Tickets and Relative Wealth. Games and Economic Behavior, 996, 4, Rosen, Sherwin Manufactured nequality. Journal of Labor Economics, 997, 5:, Skaerdas, Stergios Contest Success Functions. Economic Theory, 996, 7, We are continuing to assume that a single lottery is to be used for all layers in the contest.