# Can you incentivise competitors to handicap accurately, and also try to win?

A problem I ran into for real. A group of friends of widely differing abilities wants to hold a handicap cycling race, so that if everyone does about as well as expected, there would be a perfect dead heat. There is no time to hold qualifying round, but all the competitors have been riding together for months, so the only way to determine the handicap is to use their own knowledge of their own, and each others' abilities.

Question: is it possible to design some kind of wagering (or similar) scheme, so that:

1. each competitor is incentivised to provide the most accurate information possible, both on their ability and others' possible, with a measure of confidence (as in with betting); and
2. every competitor is still incentivised to ride as hard as possible.

I'm open to solutions that aren't strictly mathematical (eg, different kinds of things to be 'won' that aren't directly comparable), although the site may not be.

Further details In the interests of attacking a specific problem, I'll add these details:

• Competitors are familiar with the course, and have enough information to make reasonable estimates of likely times.
• The race itself will be timed, so the absolute time of each competitor can form part of any wagering, in addition to the relative finishing positions.
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Here is an attempt at an answer. Any suggestions on how to improve it are welcome, as well as any remark on flaws in the argument.

Let's formalize your setting a little bit. Suppose that

• $I$ is the set of participants.
• The racing time of any agent who rides as hard as she can is $0<t_j \sim f_j(t_j)$, where $f_j(t_j)$ is some density function.
• The handicap factor of agent $i$ is $h_i^{-1}$ ($h_i$ is a scalar) so that, given her handicap, the racing time of a competitor riding as hard as she can will be $h_j^{-1}t_j$.
• Each agent has a prior distribution on the racing time of others $p_j(t_i)$ which is the best information $j$ has on $i$'s racing abilities if $i$ was to ride as hard as she can.

So agent $j$'s strategy consists in choosing a level of effort and on providing a list of bets $b_{j1},\dots,b_{j-1},b_{j+1},\dots,$ specifying her expected time for every other competitor.

(This only allow people to reveal part of their information on others' racing time distribution. You may get better predictions from a mechanism allowing agents to reveal their distribution function, so maybe this does no meet your demand that "each competitor is incentivised to provide the most accurate information possible". It also does not satisfy your request that people be able to give information about their own abilities.)

Let each participant pay a participation fee, say $10 \$$. This is a mechanism design question, and as every mechanism design problem, it depends a lot on the solution concept your are ready to endorse and on the specification of the utility functions. One possibility is: Utility function : the agents only care about their expected pay-off -- here the money they would get out of the game -- and do not compare their pay-off to the pay-offs of others. Assume that there is no cost to effort, that is people will ride as hard as they can if it yields a higher (monetary) pay-off. Assume also that people are risk neutral. This means that, if •$j$was to bet on$i$, •$i$was to race as hard as she can, • and$j$'s reward was higher the more accurate her bet, then$j$would maximize her prior expected pay-off by betting$E_{t_i \sim p_j(t_i)} (t_i)$. Solution concept : suppose that people play according to some subgame perfect Nash equilibrium. In this setting it will mean that in the second stage of the game (the race) the agents play their best strategy for this second stage only (the strategy that maximize the additional pay-off they get from this part of the game). Then in the first stage of the game (the betting stage) they play their best strategy given that they know they will play their best strategy in the second stage. (formally a subgame-perfect Nash equilibrium only requires that both subgames be Nash equilibria but here this will imply playing a dominant strategy in both subgame so that we can simply talk of player as playing their best strategies. To be fully rigorous, one should actually consider Bayesian subgame perfect Nash equilibrium because of the distribution of types, but enough of technicalities.) Given the specification of the utility function, a way to make sure that competitors ride as hard as they can is to condition their prize-money on their time$t_j$, and the money they get from betting on the time of others only$t_{-j}$. For instance, you keep half of the participation fees, say$R$for the race prizes and you give people a fraction of$R$which is monotonically increasing in their$t_j$. An example would be giving everyone$\frac{t_j}{\sum_{i\in I} t_i} R$. With the other half$R$, you reward people's bets about others' time in a way which is monotonically increasing with the precision of the bet. For instance you could sum agents' prediction precision$\pi^j = \sum_{i\neq j \in I} |t_i - b_{ji}|$and give agents$\frac{\bar{\pi} - \pi^j}{\bar{\pi}} R$, where$\bar{\pi}$is the average precision$\sum_{i\in I} \pi^i$(assuming that$\bar{\pi} \neq 0$). Notice that, given the prize-money and betting structure, and give the utility function of the agents, at an equilibrium every agent will race as hard as she can in the second stage. Because agents cannot bet on themselves, they have nothing to loose from racing harder and a strategy profile cannot be an equilibrium (subgame perfect Nash) if they do not do so. Now at the betting stage, given that agents will race as hard as they can, the best thing people can do is to bet truthfully according to their priors, i.e. to bet$E_{t_i \sim p_j(t_i)} (t_i)$. For each agent to race as hard as possible and bet truthfully is thus the only strategy profile which meet the solution concept and you can claim that this game structure implements your requests (truthful revelation and maximal effort) in subgame perfect Nash equilibrium. From here, you need a little more work and assumptions to set the handicaps level right in order to meet your third request that the expected result of the race be as close as can be from a tie. But you see that the choice of the handicap profile has no impact on either the effort level or the revelation because people effort and prediction accuracy are rewarded based on the time profile$t_i$and not on$\frac{t_i}{h_i}$. Edit following question from the OP There might be ways to allow for self-estimation too. If you were to gain nothing from betting on yourself, you see that you would have no incentive to lie, but no strict incentive to report your true knowledge of your own capacities either. So if you want such strict incentive, it must be the case that what you can gain never from lying never outweighs what you can gain from running harder. Here is an idea on how to do so. It is imperfect, and not directly compatible with the former mechanism and assumptions, but it might give you some ideas. Assume that you know for certain a maximum time$\bar{t}$such that it is strictly impossible that anyone rides in more than$\tilde{t}$, or maybe anyone riding in more than$\tilde{t}$gets disqualifies and loses all her money. Assume that there is a finite set of possible bets$[\tilde{t},b_1,b_2,\dots,b_M]$. Agents do not know about density function this time. They only know the highest betting interval in which they are able to race, that is they know that$t_j \ in [b_s, b_{s+1}]$. This event ($t_j \ in [b_s, b_{s+1}]$) does not happen with some probability anymore. Now they can if they decide to make sure that$t_j \ in [b_s, b_{s+1}]$. Next constructs the pay-off functions as follows: • If$j$bets$b_j = \tilde{t}$and her actual time$t_j \in [\tilde{t};b_1)$she gets$a$, where$a > 0$. Otherwise, she gets zero. • Set the reward of riding in$t_j = [\tilde{t};b_1)$, at say$a$too. • Now let the reward for riding$t_i \in [b_1;b_2)$at$b > 2a$. You see that if$j$is able to race faster than$b_1$, she will never want not to only to get a better reward from her bet. Indeed, the better reward she would get would never compensate what she could have gotten by racing faster than$b_1$. • Also, let the reward of betting$b_j = b_1$when$t_i \in [b_1;b_2)$be say$b\$.

Now you could keep on this construction for all the possible betting values. You see that constructing the payoff structure in this way makes sure that racing as fast as you can is ideal (up to the interval approximation) because by racing slower will not be profitable. As you competitors will race as hard as they can, they also have an interest of betting right.

You see that this is a consequence from forgoing the probabilistic framework of the former approach. It is not straightforward to be not how to reconcile both approach. It might be possible though. Any event, I hope this simplified example gives you an idea of where to look and the kind of properties a mechanism should satisfy to allow for self-estimations.

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Thanks - this will take me a while to digest. Can you see any way to allow participants to provide information about their own abilities? In practice, I think self-estimation is worth a lot - say 2-3 times more than estimating someone else's ability. – Steve Bennett Feb 4 '14 at 1:05
You are welcome, hope this helps once digested. See my edit regarding self-estimation. – Martin Van der Linden Feb 4 '14 at 2:11

Interesting question

Maybe better to start with a inventory of all problems and solutions that can arise in this situation and then try to find a solution for each of them.

Problem 1: people will underestimate their qualities ( giving them better chance to win)

Solution 1a: you will need a system that encourages giving high capacity for yourself so a better outcome,

Solution 1b: let others set the handicap (they alll know eachother so they can make a reasonable guess of their capacties) this can lead to the opposite effect, (giving others higher capacites gives me higher chances to win) you will need a system that encourages giving high capacity for yourself so a better outcome,

Problem 2: people may underperform, if it's in their interests.

That is, if a solution to problem 1 requires people to place wagers ("I bet that I will be no faster than 18 minutes"), then they may have a stronger incentive to fulfil the bet (ie, take longer than 18 minutes) than to do their best.

Solution 2a: prevent any bet that leads to a conflict of interest (ie, I can bet on the absolute performance of others, but not on their performance relative to me, nor on my own absolute performance)

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