Caminati, Marco B
(2013)
*A contribution to an Auction Theory Toolbox through code and discussion.*
In:
39th Annual Convention of The Society for the Study of Artificial Intelligence and the Simulation of Behaviour (SSAISB).
UNSPECIFIED.
ISBN 978-1-62993-966-7

## Abstract

I am contributing a further mechanization of Vickrey’s theorem on weak equilibrium in second price auctions. Submission 6 of stage 1 is an invitation to discuss on early developments in formalizing a toolbox for auction theory, and I wish to add a further viewpoint. My formalization is in Mizar, which is nearer than other systems to the common mathematical language. Auctions being not yet present in the library, there is the chance of taking the first design decisions: one should put effort into making them suitable to build a large future library on them, and at the same time try his best to reduce as much material as possible to the most common mathematical objects, which are well supported in the existing library. I will illustrate how I faced this task. 1 What has been formalized This AISB submission provides the Mizar formalization of a theorem by Vickrey about weak equilibrium in second price auctions. The mathematics is simply and clearly exposed in [3]. A quick summary follows. The initial data is a vector b, containing the bids of each participant. Given this vector, the dynamics of the auction is simply modeled by assuming that each participant has, in his mind, a precise valuation of the auctioned good, which may or may not coincide with the amount of money he bids. The theorem in question says that, in this regard, the best strategy is to make them coincide, in the ‘weak’ sense: given a random b, changing the bid of a participant to his valuation, the payoff of that participant does not decrease. 1.1 Defining the payoff to express the theorem The best possible payoff for a single participant would be given by winning the whole auctioned good without paying anything, in which case it can be quantified by the subjective valuation v he deems the good worth. Given that, generally, any participant gets a fraction ranging from 0 (for losers) to 1 of the auctioned good, and that, of course, any decent auction scheme will impose at least to the winner(s) to disburse an amount of money, such payoff is defined by vx − p; here v is the valuation, x the fraction of the good obtained, and p the amount paid. x and p depend on all the participants’ bid, and as such each of them is a component of two distinct vectors having the same length as b; x and p are respectively termed the allocations and the payments, and are calculated from b according to the auction algorithm. Thus, the theorem’s wordy statement above can be put into this inequality: v · X II b (i)− P b (i) ≤ v · X II bi (i)− P II bi (i) , (1)