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I'd like to know if there are any examples of indirect mechanisms that are not auctions. Auctions seem like a very controlled environment. I'd like to see some examples of indirect mechanisms "out in the wild".

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Voting mechanisms would seem to be a good example. So are matching mechanisms such as those used to assign students to public schools.

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Here is an example of a mechanism "out in the wild". I take "the wild" literary, but I won't be specific because I am not a biologist and I am making this up from the top of my head.

Suppose two predators spotted some easy to get food source, and their value for the prey is their private information, i.e., a random variable $v_i$. It would be efficient to auction off the food with an appropriately designed auction. Instead the predators could fight and the one who exerts more effort in the fight wins the food as a price.

Each player $i$ maximizes his payoff $$G(e_i)v_i - e_i - \delta(e_j),$$ where $e_i$ is the effort of player $i$, and this player wins the fight with probability $G(e_i)$, the probability that he put in more effort than player $j$. The (by assumption) linear cost of effort has to be incurred independent of winning or losing the fight. On top, some damage $\delta(e_j)$ from the opponent is incurred. To solve for the optimal effort as a function of the value $v_i$, you can approach the problem like an all-pay auction.

An all-pay auction can also be interpreted as other forms of contests. It can also be used to model political lobbying activity. You can use insights from auction theory to solve for the optimal strategy in a plethora of allocation mechanisms. Think about queeuing or wars of attrition and so on.

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