1. 1 Game Theory Applications

2. Example 1: Tender Offer “Corporate Raiders” in the 1980s often used a strategy of two-tiered tender offer such as the following to take a public firm private. Suppose the current stock price for the firm is $200 and there are 500 shares. If the corporate raider obtains more than 50% of the shares (250 shares) and takes the firm private, the remaining shareholders get $180 per share. 2

3. Example 1: Tender Offer Corporate Raider makes the following tender offer to shareholders: For the first 250 shares, the corporate raider will pay a price of $210 and any share over 250, corporate raider will pay $180. For simplicity, assume all share holders make the decision of whether or not to sell to the corporate raider at the same time and if more than 250 shares are sold, which ones obtain $210 for their share is random. 3

4. If you own a share of the stock, should you sell to the Corporate Raider? 4 # of shares sold to raider by others 200 180 210 249 499 Green is expected payoff if you sell Red is expected payoff if you don’t sell 195 Expected price is 210*250/(# that sell) +180*(# that sell-250)/(# that sell)

5. If you own a share of the stock, should you sell to the Corporate Raider? 5 # of shares sold to raider by others 200 180 210 249 499 195 Dominant Strategy is to Sell

6. Example 2: Ebay bidding On-line proxy bidding where bidders enter an amount and the highest bidder obtains the item and the price paid is based on the second highest bid (second highest plus a small increment). Bidders often wait to the last possible instant to bid (called sniping) so there is little feedback about other bids at the time an individual places a bid. 6

7. Example 2: Model of Ebay You know your valuation but know little about other bidders. All bidders choose bid simultaneously. Highest bidder gets the item and pays the second highest bid. Referred to as a Second-Price Sealed Bid Auction 7

8. Example 2: Ebay Game RdValuationBidPayoff 1 2 3 4 5 6 AVE. RdValuationBidPayoff 1 2 3 4 5 6 AVE 8 TEAM 1TEAM 2 http://www.random.org/

9. Strategy in a Second Price Sealed Bid Auction when Bidders Valuation is V=100 9 Highest Rival Bid V=100 B=90 What if Bid, B, which is less than V? For example, B=90. Payoff

10. Strategy in a Second Price Sealed Bid Auction when Bidders Valuation is V=100 10 Highest Rival Bid V=100 B=110 What if Bid, B, which is more than V? For Example, B=110 Payoff

11. Strategy in a Second Price Sealed Bid Auction when Bidders Valuation is V=100 11 Highest Rival Bid V=100 V=B=100 What if Bid, B, which is equal to V? Say V=100 so B=100 Payoff

12. Comparison Of Different Strategies When B=V compared to B<V  When another bidder bids between B and V (90 and 100), Bidder wins if B=V and obtains a positive payoff but losses when B<V.  When another bidder does not bid between B and V, then payoff is the same for both strategies. When B=V compared to B>V  When another bidder bids between V and B (100 and 110), Bidder losses if B=V and wins when B>V. The payoff of winning is negative.  When another bidder does not bid between V and B, then payoff is the same for both strategies. 12

13. (Weakly) Dominant Strategy In a Second Price Sealed Bid Auction, the bidder does at least as well if she bids her valuation (B = V) and strictly better in some cases than if she bids any other value. Therefore, bidding her valuation is a weakly dominant strategy. 13

14. 2 nd Price Sealed Bid Auction and English Auction An English (or open outcry) auction is one where bidders shout bids publicly. Auction ends when there are no higher bids. Termed a “button auction” in Japan. (like Sotheby does) Strategies are the same in both auctions. In English auction, bidder should drop out when bid is greater than valuation. They are strategically equivalent. 14 http://www.bing.com/videos/search?q=30+for+30+naismith+rules&FORM=VIR E1#view=detail&mid=94718A50722BF95144FB94718A50722BF95144FB http://espn.go.com/video/clip?id=8395320

15. Example 3: 1 st Price Sealed Bid Auction An auction where bidders submit one bid in a concealed fashion. The submitted bids are then compared and the person with the highest bid wins the award, and pays the amount of his bid. (ex. refinancing credit, foreign exchange, gov’t contracts, mining leases) Strategically equivalent to a Dutch Auction where the auctioneer begins with a high asking price which is lowered until some participant is willing to accept the auctioneer's price. 15

16. Example 3: 1 st Price Sealed Bid Auction Game RdValuationBidPayoff 1 2 3 4 5 6 AVE. RdValuationBidPayoff 1 2 3 4 5 6 AVE 16 TEAM 1TEAM 2 http://www.random.org/

17. How Would Things Change if there were Three Bidders? In 2 nd price sealed bid auction, weakly dominant strategy is still for players to bid their valuations. Expected revenue from auctioneer would increase because second highest bid likely to be more. In 1st price sealed bid auction, players would shade their bids downward less when the number of bidders increases. Expected revenue from auctioneer would increase (the same amount as in the 2 nd price sealed bid auction). 17

18. What Auction to Use? The Second Price Sealed Bid Auction, the First Price Sealed Bid Auction, the Dutch Auction and the English Auction are REVENUE EQUIVALENT. This means that the expected revenue generated is the same from these four auctions. Punchline: As an auctioneer, rules of the auction does not affect revenue much but reservation price and “ease of entry” do. 18

19. Example 4: Right of First Refusal Many contracts contain a Right of First Refusal clause – including the collective bargaining agreement in the NBA pertaining to restricted free agents. http://www.nbpa.com/cba/2005 Article XI, Section 5 allows an incumbent firm (current team) to retain the right of an employee (player) by matching the best offer made to that employee (player) by a rival firm (another team). 19

20. ESPN.com on July 17, 2014 According to ESPN's Chris Broussard, the Phoenix Suns and Eric Bledsoe won't be agreeing to terms on a new contract any time soon: The future of one of the most talented free agents left on the market remains cloudy as Eric Bledsoe and the Phoenix Suns remain far apart in contract talks, according to sources close to the situation. Bledsoe's representatives have been engaged in discussions with Phoenix, but the Suns' offer is far below what Bledsoe is looking for. And while Bledsoe remains one of the premier names available from a loaded class of free agents, Broussard notes a resolution may take some time to manifest itself, for rival teams are wary of extending offer sheets to the 24-year-old. Per Broussard, "the belief that the Suns will match any offer Bledsoe receives from a competing club has deterred teams from aggressively pursuing the 24-year-old point guard." 20

21. Example 4: Right of First Refusal Suppose you are one of three players in the negotiations process: current team, player and rival team. The contract of the player is about to expire and the current team is deciding what offer to make the player. The player is worth $10M to the current team (based on their next best alterative). The current team makes an initial offer to the player (O I1 ) which the player can accept or initially reject. If the player rejects O I1, the rival team decides whether to make an offer (O R ) or not make an offer. The rival team incurs a cost of $.5M when putting together an offer. The player is worth $10M to the rival team. If the rival team makes an offer, the current team can match this offer or not (actually the current team can make any offer they want, O I2 ). The player then decides whether to accept the current team’s offer, the rival team’s offer or reject both. If the player does not agree to a contract, the player must sit out the year and his payoff would be zero. 21

22. 22 I R P P R P I O I1 OROR O I2 Accept O R Reject O I1 Accept O I1 Initially Reject O I1 Accept O I2 I, incumbent firm P, player R, rival firm Payoffs (incumbent, player, rival) Reject (10-O I1, O I1,0) (0, O R,10-O R -.5) (10-O I1, O I1,0) (0,0,0) (0,0,-.5) (10-O I2, O I2,-.5) No Offer Make Offer RIGHT OF FIRST REFUSAL

23. Subgame Perfect – Backward Induction Player: Accept O R if O R >O 12 and O R >0 Accept O l2 if O 12 >O R and O l2 >0 Reject if O R <0 and O l2 <0 Incumbent:Offer O l2 =O R if O R <10 Offer O l2 10 Rival Firm:If make offer, offer O R <10 Rival Firm:No Offer Player (if rival firm makes no offer):Accept O l1 if O l1 >0 Player:Accept O l1 if O l1 >0 Reject if O l1 <0 Incumbent: Offer O l1 =0 Outcome – Incumbent offers 0 and Player Accepts 23

24. What if there was not a Right of First Refusal? Suppose the game is the same as before but now, if the player rejects O I1, and the rival firm makes an offer O R, the incumbent firm cannot make another offer. What is the likely outcome to this game? 24

25. 25 I R P P R P O I1 OROR Reject O I1 Accept O I1 I, incumbent firm P, player R, rival firm Payoffs (incumbent, player, rival) Reject (10-O I1, O I1,0) (0, O R,10-O R -.5) (10-O I1, O I1,0) (0,0,0) (0,0,-.5)(10-O I2, O I2,-.5) No Offer Make Offer NO RIGHT OF FIRST REFUSAL Accept O R Accept O I1 Initially Reject O I1

26. Subgame Perfect – Backward Induction Player: Accept O R if O R >O 11 and O R >0 Accept O l1 if O l1 >O R and O l1 >0 Reject if O R <0 and O l1 <0 Rival Firm: Offer O R =O l1 if O l1 <9.5 Offer O R 9.5 Rival Firm:Make Offer if O l1 <9.5 No Offer if O l1 >9.5 Player (if rival firm makes no offer):Accept O l1 if O l1 >0 Player: Accept O l1 if O l1 >9.5 Reject O l1 if O l1 <9.5 Incumbent: Offer O l1 =9.5 Outcome – Incumbent offers 9.5 and Player Accepts 26

27. Strategy Depends on Sophistication of Other Players Rick Reilly article on Rock, Paper, Scissors 27

28. Strategy Depends on Sophistication of Other Players Liars Poker: http://wizardofodds.com/games/liars-poker/ http://quant.stackexchange.com/questions/4201/strategies-for-liars-poker http://www.youtube.com/watch?v=AIxt2pJx31o 28

29. Strategy Depends on Your Information and what you know about Other Player’s Information Alice and Bob are waiting for an interview with a potential employer, each wearing a blue hat whose color they have forgotten in the hustle of getting ready. The receptionist announces that either or both of them may proceed to the interview if they know the color of their hat (while admonishing them not to look at their hats and not to speak). Alice and Bob continue to sit, each looking at the other's blue hat while paralyzed by uncertainty as to the color of their own. After some time, the receptionist remarks that one of their hats is blue. WHAT HAPPENS NEXT? 29 Lesson: In most games a player benefits from having more information if other players in the game do not know he/she has the information. Information can be detrimental if other players know you have the information.