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Share Auctions, Pre- Communications, and Simulation with Petri-Nets Research Team: Anna Ye Du (SUNY, Buffalo), Xianjun Geng (UW, Seattle), Ram Gopal (UC, Storrs), Ram Ramesh (SUNY,Buffalo), Andrew B. Whinston (UT, Austin)

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What Are Share Auctions Single indivisible item auction Share auction Bidder 1 $100 Bidder 2 $150 Bidder 3 $200

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Bid Schedule In Share Auctions price p shares x20%40%60%80%100% Bid schedules bidder 1 bidder 2 bidder 3

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Examples Of Share Auctions Treasury securities (Wilson 1979) 3G radio spectrum (Klemperer 2004) Dedicated WAN circuits (Approximately) A large number of identical items (ubid.com)

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The Issue of Low Market Clearing Prices In Share Auctions (Wilson 1979) “ (A) share auction can yield a significantly lower sale price. ” Collusion as a reason for low profits (McAfee and McMillan 1992) Bidding rings (Klemperer 2004) Empirical evidence

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Our Research Focus: Understanding Bidder Collusion In Share Auctions Multiple equilibria in a linear demand model, and tacit collusion The case of two bidding rings, and explicit collusion through pre-communication Simulation of collusion using Petri-Nets

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Our model extends Wilson 1979 n 2 bidders Identical valuation V (common and known value) Symmetric bid schedule x(p) Wilson shows one equilibrium strategy is to submit a bid schedule x(p)=(1-2p/(nV))/(n-1) Market clearing price p*=V/2 Seller receives only half of bidder valuation Any other possible market clearing prices? A Linear Demand Model

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A Linear Demand Model (Cont.) We consider the class of linear bid schedules Demand function takes form x(p)=ap+b Given any a<0, there is a symmetric equilibrium where The optimal bid schedule is The market clearing price is

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Multiple Equilibria and Tacit Collusion Multiple equilibria exists (by varying a) Depending on a, market clearing price can be anywhere between 0 and V Recall p* V when a - ; p* 0 when a -1/(n(n-1)V) If bidders can collude on a larger a, they can force a lower market clearing price However, little is know on how this tacit collusion can happen

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Pre-Communication and Explicit Collusion Notice that when n , p* V Bidders have incentives to merge and shrink the market before bidding A two-stage model of explicit collusion 1 st stage: pre-communication to form bidding rings 2 nd stage: bidding rings make collusive bids “ Collusive bids ” means same a ’ s (i.e. same bid schedules) that are negotiated via pre-communication

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Pre-Communication n bidders are divided into two rings One with k bidders, and the other with n-k bidders Alternatively, only one ring with k bidders, and all other bidders independent Bidders in a ring coordinates with each other Question: do the k bidders have incentives to form a ring to deviate from the above bid schedule x*(p)?

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Pre-Communication and Bidding The k buyers want to deviate and construct a new price schedule x k (p)=(a+ k )(p-V)+(n-2)/n(n-1)+ k, where k and k are deviation parameters. Similarly, in equilibrium the n-k buyers know the above deviation and make a response by changing their price schedule to x n-k (p)=(a+ n-k )(p-V)+(n-2)/n(n-1)+ n-k The deviation pair (( * k, * k ); ( * n-k, * n-k )) is a Nash equilibrium (NE), if simultaneously, the k bidders and the n-k bidders Maximize each of their profit Subject to the market clearing constraint and Subject to the constraint of the reserve price

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Pre-Communication and Bidding (Cont.) The k bidders solve the problem Subject to market clearing constraint Subject to reserved price constraint a+ k The n - k bidders solve the problem Subject to market clearing constraint Subject to reserved price constraint a+ n-k

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Findings About Explicit Collusion Finding 1 (Stability of allocation x*(p*)) : In equilibrium, x k *(p) and x n-k *(p) cross at the market clearing price, p*. Bidding rings do not affect share allocation, even if k n/2 Open question: does it hold when #rings > 2? Finding 2 (Instability of market-clearing price p*): The two bidding rings will cooperatively drive the market clearing price down as low as allowed by the seller ’ s reserve price In equilibrium bidder profit depends only on p*, which decreases with * k and * n-k.

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Factors Affecting Collusion In Repeated Interactions Repeated share auctions are most popular Various collusive factors cited in prior literature (Vives 1999) The number of firms ( n ) Time lag between two auctions (Time Lag) Frequency of prior auction experience (Auction Times) Frequency of interaction among bidders (Interaction) Weight of the future Multi-market contact …

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A Heuristic Decision-Making Model - Overview n Time Lag Auction Times Interaction Price % of Ring Participation Number of rings

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A Heuristic Decision-Making Model - Assumptions On entering 2 nd stage, each bidder has complete information of her own bidding ring. Lacking outside information, a bidder in a ring of size k believes that the expected value of the size of all the other rings can be estimated by the value k. A bidder ’ s belief in gain from forming a ring is A bidder ’ s belief in gain from forming a ring of size k price without rings (i.e. bid by n individual bidders) price with rings all of size k (i.e. bid by n/k rings, each of which is a group of k bidders) = - i.e.

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A Bidder ’ s Dynamic Decision Tree

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Simulation with Petri Nets

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Bidders Search a Satisfactory Ring

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Simulation Results n Time Lag Auction Times Interaction Price % of Ring Participation Number of Rings -0.00065 0.00043 0.01880 0.13380 -0.00207 -0.00016 0.00062 0.13301 -0.89875 1.01884 -0.00098 The solid arrows are significant at 0.01 level. The dash arrows are not significant at 0.05 level. 5 relations (in green) in the model are confirmed, and 2 (in yellow) are not.

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Conclusions We construct a linear demand model that shows the multiplicity of equilibria in complete-information common-value share auctions Bidders can get any market clearing price through tacit collusion Reducing the number of bidders help remaining bidders We model a two-stage game to incorporate pre-communications for explicit collusion In the two rings setup we show that rings do not affect share allocation Rings lead to a lower market clearing price We construct a heuristic decision-making model for repeated share auctions with bidding rings. Simulation in Petri Nets shows: Price increases with #rings, decreases with N and ring participation

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