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)

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

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

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)

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

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

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

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

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

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

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)?

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

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 

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.

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 …

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

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.

A Bidder ’ s Dynamic Decision Tree

Simulation with Petri Nets

Bidders Search a Satisfactory Ring

Simulation Results n Time Lag Auction Times Interaction Price % of Ring Participation Number of Rings 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.

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