Tag Archives: MDV3100

Maldovanu and Sela designed a competition model with elimination

Maldovanu and Sela (2006) designed a MDV3100 model with elimination stages. The effort\’s cost function includes an ability parameter that is private information for each competitor. The authors show that when the cost is a concave (or linear) function of effort, the equilibrium total expected effort in a single contest with a unique prize is greater than in a competition scheme with elimination stages where in each stage a certain number of prizes are allocated.

The model
The model encompasses an all-pay auction of an indivisible object, with N≥2 risk-neutral participants who have the following characteristics:
Define the random variable X as follows:
so that p=Pr[X=1]. Assume that individual i has a Bernoulli utility defined as:
where with ω>0. His expected utility in a lottery p will be:

For any bidder i, the winning probability function is defined as , where b is a bid of agent i. It is implicit that the winning probability is anonymous, depending solely on the participant\’s bid. The assumption of anonymity may not be supported empirically. Castelar et al. (2009), for example, analyzed a specific public contest and concluded that there are socio-economic characteristics of individuals that interfere dramatically on the winning probability. In other words, with the same effort (bid) individuals may have different winning probabilities. On the other hand, if this supposition is withdrawn a symmetric equilibrium would not be possible, not to mention that anonymity is among the desired axioms of success functions in contests.

In Assumption 1ɛ is an exogenous probability of winning that will be called from now on as “luck factor”. The assumption that p(·) is strictly increasing guarantees a first-order stochastic dominance for the lotteries that are associated with higher bids. Furthermore, Supposition 3 (allow the possibility that exist) guarantees the existence of a pool of sufficiently low types that rely on luck. Based on this possibility, Definition 1, makes a distinction between bidders in the Chinese auction.

For any participant i a strategy is a function such that β(0)=0 and . It is assumed that β is a continuous and increasing function. In a symmetric equilibrium it is admitted that β=β for any i. It is worth mentioning that, implicitly, even with symmetric bidders there is a possibility of pooling for those below .
The nature of competition requires that each effective competitor submits his bid in such a way that his expected utility is maximized, considering that he or she expects to have the maximum winning probability in the lottery that is assigned according to his bid. An effective competitor is only concerned with competition against other effective rivals, since the others (if any) receive a lottery with an exogenous winning probability ɛ that is common knowledge. Therefore, obtaining the maximum winning probability should be conditioned to competition solely against other effective competitors.
It is assumed that each effective competitor is uncertain about the number of existing effective rivals and, therefore, his expected gain should include this uncertainty. Hence, being an effective rival or not is modeled as a binomial random variable whose probability of success (being an effective rival) is common knowledge since .
From the perspective of a give bidder, if the number of effective rivals is a random variable , then the probability that there are exactly K effective rivals is given by

The possible values of lie between 0 (there are no effective rivals) and N−1 (all other participants are effective rivals).

Equilibrium
Let a participant i with and define the set such that . Hence, if i believes that there are exactly K effective rivals, his ex-ante expected utility could be written as follows:

Since p(·) and β(·) are increasing functions, (1) can be rewritten as
where e are, respectively, the highest and the lowest order statistics amongst K independent random variables. Given i\’s uncertainty towards the number of effective rivals, he chooses his bid so that