Michal.Bresky@Michal.Bresky@cergecerge-ei.-ei.czcz

Michal BresMichal Breskyky

(Summer 2007)

Existence of Pure Existence of Pure Equilibria in Uniform Price Equilibria in Uniform Price

Multiple-unit Auctions Multiple-unit Auctions with Private-value with Private-value

BiddersBidders

Existence of Pure Equilibria in Uniform Price Multiple-unit Auctions with Private-

value Bidders

Literature:• Dasgupta and Maskin (1986) RES, Reny (1996) Econometrica, Simon and Zame (1990)

Econometrica, Jackson and Swinkels (1999) Econometrica.• Amman and Leininger (1996) GEB, Krishna and Morgan (1997) JET,

Engelbrecht-Wiggans and Kahn (2002) ET, Ausubel and Cramton (2005) ET, Back and Zender (1993) RFS, Noussair (1995) ET.

Results:• Equilibrium in multi-unit auctions exists.• Every equilibrium in auctions can be rearranged to pure weakly increasing one.

Technique:

• The shape of the payoff function when bids are equal (and some but not all are winning) - ties - the only source of discontinuities.

• In equilibrium no tie occurs (with positive probability) because every bidder is typically strictly better off if he bids slightly above the tie instead of on the tie.

• Therefore, when searching for an equilibrium one can a priori eliminate all player profiles in which ties occur with positive probability.

• Then the seller can choose any rule to break the ties with no influence on the set of bidder equilibrium strategies.

• Then the existence theorem by Reny (1999) applied for one specific tie-breaking rule guarantees the existence of equilibrium with any tie-breaking rule (e.g. the "random" rule that is usually considered the literature).

THE GAME

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. where];...,,[ valuessBidder' 1,,,1, jijikiii vvvvv

supportcompact on the valuesof PDF the- )( ii vF

identical units of goods for sale.

bidders in an auction.

Independence is required between a pair of values of two different bidders (not between values of one bidder)

bids submitsbidder Each k

.

wins. above bidshighest Rk

Vi j 1l 0,v j

bi,1 , ,bi,k 1, B

Features:

•

• Examples: Vickrey auction, Pay-your-bid auction, Uniform-price auction, Dutch Auction, All-pay auction, and linear combination of these auctions.

• There is a bound such that no bidder has an incentive to bid above his value instead of bidding the value or below, In addition, in any tie below the value than the bidder prefers (strictly) to win the tie than to lose it.

THE GAME

.

.

A tie occurs when kth and k-1st highest bids are equal (and other bids may be tied with them).

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Ties and Tie-Breaking Rule Equivalence

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A tie occurs when kth and k-1st highest bids are equal (and other bids may be tied with them).

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GG

TMGTMG Niii

Niii 1. Theorem

Ties and Tie-Breaking Rule Equivalence

Example - equilibrium does not exists with discontinuous distributions and random tie-breaking:

• Bidder 1 has value = 1 with probability 1.• Bidder 2 has value distributed uniformly on [1,2].• Claim: There is no equilibrium.• To any first bidder pure strategy the second bidder does not have best response.• Efficient tie-breaking rule breaks the ties in the favor of the second bidder equilibrium

exists • (but not with random tie-breaking).

• bouunded from above,

• sum upperr-semi continuous, and

• payoff secure,

then equilibrium exists by Reny (1999).

Equilibrium Existence in Game with Resricted Strategy Space

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Equilibrium Existence in Game with Resricted Strategy Space

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Equilibrium Existence in Game with Resricted Strategy Space

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PURE WEAKLY INCREASING EQUILIBRIA

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Thank you for your attantion.