Parham Holakouee
PHDBA 279B
Spring 2012
Cognitive Biases and Contestant Over-Exertion
I. Introduction
Cognitive biases can induce deviations from rational strategic behavior in contests. I am interested in exploring the implications of contest structures in which predictable deviations from rationality are pervasive and systematic. Equilibrium outcomes in the contest theoretical literature routinely assume players’ probabilistic expectations of all relevant information equal the true probabilities. However, experimental research – and some empirical observations -- demonstrates over-dissipation by contest participants with effort levels exceeding the predicted equilibrium. Incorporating predictable deviations from rationality into the theoretical models may provide some insight regarding a possible source of the discrepancy between equilibrium predictions and experimental observations.
II. Employment Tournament
In an employment tournament context, the employment contract can be analyzed as a contest designed by the employer to maximize its expected payoff. The employer will compensate its employees via a tournament, in which payoffs are contingent on relative performance if arrangement maximizes expected employer rents.
In this tournament context, the employer (“Organizer”) will set the rules of the tournament and its employees (“Contestants”) will choose effort levels in response to these contest parameters. We will assume that the Contestants are vying for a single prize that is set equal to the net present value of earning partnership status within the firm.
I am interested in analyzing how the strategy of the Organizer and Contestants is influenced by a systematic increase in each Contestant’s subjective probability of winning the contest while the true probability of winning remains unchanged. While the subjects carry a subjective, irrational belief regarding their likelihood of winning, the Organizer is aware of both the true probability of winning and of the Contestants’ bias.
A. Excessive Effort
The experimental literature on contests reveals that individuals frequently exceed the effort levels predicted by theoretical equilibrium analysis (Davis and Reilly, 1998). These results are even more pronounced in contests with noise. The experimental literature and empirical observations of effort levels exceeding equilibrium predictions can possibly be reconciled with the theoretical literature if we take account of systematic cognitive biases among Contestants. Specifically, overconfidence can be incorporated into the models to account for excessive effort levels corresponding to subjective perceptions of the probability of winning that exceed the true probability.
1. Overconfidence
There is substantial psychological research demonstrating the pervasive tendency to overestimate the likelihood of success relative to one’s peers. Alpert and Raiffa (1982), Buehler, Griffin, and Ross (1994), Weinstein (1980) and Kunda (1987) find that people believe good things happen more often to them than to their peers. Langer and Roth (1975), Weinstein (1980) and Taylor and Brown (1988) find that people are overly optimistic about their own ability as compared to others.
2. Overconfidence Among High Achievers
There is reason to believe that this over-optimism may be even more pronounced among certain populations. It could be illuminating to focus our attention on tournaments in which the pool of prospective Contestants and those among the pool selecting into the tournament yield Contestants with overconfidence levels exceeding that in the general population. Specifically, graduates of top ranked graduate schools earning positions in the most selective investment banks, consulting firms, and law firms would be expected to place at the high end of the overconfidence distribution.
The prospective candidates to these sought-after positions have been consistently successful relative to their peers as indicated by their ability to earn a position at a top-ranked graduate program. To be considered for a position in a prestigious firm, they have likely excelled even when competing with high-caliber peers. Moreover, they have likely self-selected into these more competitive endeavors and thus have a strong belief in their ability to compete and win. In addition, by selecting into a competitive position in a firm in which only the top performers are promoted to higher positions, they are once again demonstrating their high level of competitiveness, and presumably, overconfidence.
Recent hires at a top-ranked financial, law, or consulting firm are likely to be particularly susceptible to falling prey to the bias of overconfidence. Being accustomed to consistently achieving all goals they have set their mind to, they are likely to maintain a firm conviction that they will achieve their goal of being a star within this new employment context as well.
B. Firm Rent Extraction Due to Employee Overconfidence
If indeed this group is systematically overconfident, firms (contest Organizers) may have an incentive to utilize that misperception to extract rents from these budding superstars. The firm, as contest Organizer, can benefit from this overconfidence by providing a salary largely contingent on relative achievement. If the Contestants’ average subjective belief regarding the likelihood of winning the tournament exceeds the true probability – and the firm is aware of both the true probability and of this systematic deviation of the subjective probabilities – the Organizer can extract rents by providing compensation contingent on winning the contest.
This is somewhat analogous to the lottery contest discussed in Section III. However, the primary source of divergence between the subjective and true probability of winning is not innumeracy but the overconfidence of the Contestants.[1] In this context, even if the Contestant has full and accurate information regarding the objective probability of winning, their overconfidence leads them to believe they have a higher probability of winning than the true, objective probability for a given Contestant.
C. Example: Law School Graduates
For a concrete example of this contest, we can analyze the behavior of recent law school graduates working as 1st year junior associates at a large law firm (this is an example I was (un)fortunate enough to experience first-hand and so have some familiarity with). For purposes of this contest analysis, we can break down the total salary to the associate as Current Salary and Tournament Salary. The Current Salary is simply an amount the employer pays each associate for work in the current period. The Tournament Salary is equal to the net current period value of the expected salary of becoming a partner at the firm for the expected period a lawyer remains a partner minus the anticipated outside option to the associate for this period (we assume the outside option is always less than the partner salary and uniform across all Contestants). After determining the total compensation, the firm and associate decide on the optimal allocation between Current Salary and Tournament Salary. In this contest, the players are competing only with respect to the tournament component of the salary.
The prize B can be assumed to incorporate all non-monetary rewards of being selected as a partner at the law firm (non-pecuniary value of winning, prestige, etc.). B is equal to the present value of all net expected returns from becoming a partner.
1. True Probability Versus Subjective Probability
A further assumption we will make with respect to the Tournament Salary is that the firm has an accurate perception of each associate’s probability of earning a partnership position: (T)Pr. This assumption is in accord with our setup since we presume the firm has ex ante determined that exactly one associate out of the n associates will be selected as a partner.[2] Whether this probability for any individual Contestant i fluctuates over time from the perspective of the firm is not critical to the analysis. The firm knows it will award one prize to the n associates; its expected tournament payment, B, is fixed.
However, the associates have subjective beliefs regarding their expected probability of winning which deviate from the true probability; the Contestants’ belief diverges with that of the firm regarding the monetary value of the tournament component of the salary. We will assume that, due to the overconfidence of the associates, each associate i’s subjective probability of winning exceeds the actual probability. This higher probability is attributed to the Contestant i believing every unit of her effort is worth double the effort of that of all other players j ≠ i. We will assume that each associate i is unaware of the symmetric “bloated” expected tournament salary of her competitors j ≠ i. We will also assume that each Contestant i believes all other players j ≠ i do not know i believes her effort is worth more than that of the other players. We will assume further that this higher IA(Pr) of the associates is symmetric across each player i.
Associates will prefer to receive a greater proportion of their salary as Tournament Salary since the probability weight associate i places on her probability of winning exceeds the firm’s expected probability that associate i will win the tournament: (IA)Pri > (T)Pr for all i. Analogously, firms will prefer to compensate associates with Tournament Salary since the expected payout from firm to associate is lower with Tournament Salary relative to Current Salary when associates hold subjective probabilities of winning exceeding the true probabilities.
D. Employment Tournament Model with Overconfidence
We can set the parameters of this employment tournament contest as follows:
• Contestants N = {1, 2, 3, 4}
• Cost of Effort: Ci (xi) = xi
• Each Contestant has the same cost function equal to the total units of effort expended.
• These effort units, xi, are equal to each dollar of Current Salary sacrificed in exchange for Tournament Salary (described below).
• Prize = B
• One Contestant will receive the entire prize.
• Each Contestant i has the same belief regarding the value of the prize.
• The Organizer (firm) also has the same valuation of the prize.
• There is a constraint on effort, m, and m < ¼(B). Therefore, every player will exert effort equal to this maximum value m (Che and Gale 1998).
• Total effort in this contest is fixed at 4m. The “effort” component of the contest is the decision by Contestants to opt for Tournament Salary in place of Current Salary.
• All Contestants will choose to transfer the maximum amount m of Tournament Salary in lieu of Current Salary as long as the expected payoff exceeds that from Current Salary.
• Each Contestant i is overconfident about the relative effect of her total effort relative to her competitors. Therefore, Contestant i believes xi = 2xj for all j ≠ i. Each Contestant holds this belief regarding the relative impact of her effort compared to her competitors.
• Every Contestant i believes all Contestants j ≠ i believe the effect of their effort xj is equal to that of all other players.
• Every Contestant i believes all other Contestants j ≠ i do not know i believes her effort is worth more than that of the other players.
• Therefore, Contestants’ uniform Irrational Anticipated belief regarding the probability of winning is: (IA)Pri = 2xi / Σj=1 to n xj, j = all players [1, …, n].
• (IA)Pri = 2m / 5m
• (IA)Pri = 2/5 (0.4)
• True probability of winning for each Contestant i: (T)Pr = xi / Σj=1 to n xj
• (T)Pri = m / 4m
• (T)Pr = 1/4 (0.25)
• Contestants are risk-neutral.
• Compensation from firm to employee can be paid as Current Salary or Tournament Salary. For every $1 of Current Salary, employee can instead opt for an amount $3.50 in Tournament Salary. The present value of this amount is $3.50 multiplied by the probability of winning the tournament.
• The Tournament Salary yields a higher “irrational” expected payoff to a risk-neutral Contestant than Current Salary as long as the Tournament Exchange Rate is greater than $2.50.
• A Tournament Salary Exchange Rate lower than $4 yields a higher true expected payoff to a risk-neutral Organizer.
• Within the Tournament Exchange Rate ranging [$2.50, $4], Contestant(s) and Organizer are willing to exchange Current Salary for Tournament Salary.
• While Contestants yield a higher irrational expected payoff, exchanging Current Salary for Tournament Salary at any price lower than $4 yields a negative true expected payoff to Contestants.
• Organizer earns rents from Contestants at any Exchange Rate greater than $2.50.
• With risk-neutral Contestants, Organizer should be able to implement a Tournament Exchange Rate equal to $2.50 leaving Contestants indifferent between Tournament Salary and Current Salary.
• If we: (1) relax the risk-neutrality assumption or (2) lower the degree of overconfidence below the 2 to 1 ratio, Tournament Salary in lieu of Current Salary can still yield rents to the Organizer at any Exchange Rate below $4.
III. Lottery Contest
State lotteries provide an appropriate context to study more subtle sources of deviations from rationality in contests: basic innumeracy and the tendency to overweight very small probabilities.
The state lottery context illustrates how these predicable biases can yield rents to the contest Organizer. In this contest, Contestants have an extremely small probability of winning a very large prize. Since a defining feature of this form of lottery is that total Contestant efforts outweigh total prizes, the lottery contest is always a negative expectation game from the perspective of the Contestants (and positive expectation for the Organizer). For purposes of clarity, a unit of "effort" here is set equal to the purchase of a single ticket. The probability of winning with a single ticket multiplied by the prize(s) is always less than the cost of a single ticket.
A. Innumeracy
There is experimental and empirical evidence to indicate that individuals are limited in their ability to understand the mathematical implications of extremely small probabilities. Consequently, we may expect some level of pervasive irrationality when it comes to implementing a strategy choice in the face of such extreme probabilities. Distinguishing, for example, between a 1x10^(-5) probability of receiving a prize and the identical prize but with a 1x10^(-10) probability of winning should induce a substantially different optimal strategy. However, individuals are frequently incapable of appropriately distinguishing between the expected value implications of these two probabilities – they are often both placed under the broad category of “unlikely events” and there is no meaningful change in strategy when facing one highly improbable event relative to another.