A Note on k-price Auctions with Complete Information When Mixed Strategies are Allowed

Restricting attention to players who use pure strategies, Tauman (2002) proves that in a k-price auction (k\u3e 3) for every Nash equilibrium in which no player uses a weakly dominated strategy: (i) the bidder with the highest value wins the auction and (ii) pays a price higher than the second-highest value among the players, thereby generating more revenue for the seller than would occur in a first- or second-price auction. We show that these results do not necessarily hold when mixed strategies are allowed. In particular, we construct an equilibrium for k \u3e 4 in which the second-highest valued player wins the auction and makes an expected payment strictly less than her value. This equilibrium–which exists for any generic draw of player valuations–involves only one player using a nondegenerate mixed strategy, for which the amount of mixing can be made arbitrarily small

kth price auctions and Catalan numbers

This paper establishes an interesting link between kth price auctions and Catalan numbers by showing that for distributions that have linear density, the bid function at any symmetric, increasing equilibrium of a kth price auction (k is 3 or higher) can be represented as a finite series of k-2 terms whose lth term involves the lth Catalan number. Using an integral representation of Catalan numbers, together with some classical combinatorial identities, we derive the closed form of the unique symmetric, increasing equilibrium of a kth price auction for a non-uniform distribution

kth price auctions and Catalan numbers

This paper establishes an interesting link between kth price auctions and Catalan numbers by showing that for distributions that have linear density, the bid function at any symmetric, increasing equilibrium of a kth price auction (k is 3 or higher) can be represented as a finite series of k-2 terms whose lth term involves the lth Catalan number. Using an integral representation of Catalan numbers, together with some classical combinatorial identities, we derive the closed form of the unique symmetric, increasing equilibrium of a kth price auction for a non-uniform distribution