Game Theory.

Game Theory

Applications of Game Theory
Firm Interaction National Defense – Terrorism and Cold War Auctions Sports – Cards, Cycling, and race car driving Politics – positions taken and $$/time spent on campaigning Parenting / Nanny Monitoring Traffic / Road Planning Legal System Group of Birds Feeding

Game Theory Terminology
Simultaneous Move Game – Game in which each player makes decisions without knowledge of the other players’ decisions (ex. Cournot or Bertrand Oligopoly). Sequential Move Game – Game in which one player makes a move after observing the other player’s move (ex. Stackelberg Oligopoly).

Game Theory Terminology
Strategy – In game theory, a decision rule that describes the actions a player will take at each decision point. Normal Form Game – A representation of a game indicating the players, their possible strategies, and the payoffs resulting from alternative strategies.

Example 1: Prisoner’s Dilemma (Normal Form of Simultaneous Move Game)
Martha’s options Don’t Confess Confess Peter’s Options M: 2 years P: 2 years M: 1 year P: 10 years M: 10 years P: 1 year M: 6 years P: 6 years What is Peter’s best option if Martha doesn’t confess? Confess (1<2) Confess (6<10) What is Peter’s best option if Martha confess?

Example 1: Prisoner’s Dilemma
Martha’s options Don’t Confess Confess Peter’s Options M: 2 years P: 2 years M: 1 year P: 10 years M: 10 years P: 1 year M: 6 years P: 6 years What is Martha’s best option if Peter doesn’t confess? Confess (1<2) Confess (6<10) What is Martha’s best option if Peter Confesses?

Example 1: Prisoner’s Dilemma
Martha’s options Don’t Confess Confess Peter’s Options 2 years , 2 years 10 years , 1 year 1 year , 10 years 6 years , 6 years First Payoff in each “Box” is Row Player’s Payoff . Dominant Strategy – A strategy that results in the highest payoff to a player regardless of the opponent’s action.

Example 2: Price Setting Game
Firm B’s options Low Price High Price Firm A’s Options 0 , 0 50 , -10 -10 , 50 10 , 10 Is there a dominant strategy for Firm B? Low Price Is there a dominant strategy for Firm A? Low Price

Nash Equilibrium A condition describing a set of strategies in which no player can improve her payoff by unilaterally changing her own strategy, given the other player’s strategy. (Every player is doing the best they possibly can given the other player’s strategy.)

Example 1: Nash? Nash Equilibrium: (Confess, Confess) Martha’s options
Don’t Confess Confess Peter’s Options 2 years , 2 years 10 years , 1 year 1 year , 10 years 6 years , 6 years Nash Equilibrium: (Confess, Confess)

Example 2: Nash? Nash Equilibrium: (Low Price, Low Price)
Firm B’s options Low Price High Price Firm A’s Options 0 , 0 50 , -10 -10 , 50 10 , 10 Nash Equilibrium: (Low Price, Low Price)

Chump, Chump, Chump

Game Theory and Politics
Game Theory for Swingers: What states should the candidates visit before Election Day? Some campaign decisions are easy, even near the finish of a deadlocked race. Bush won't be making campaign stops in Maryland, and Kerry won't be running ads in Montana. The hot venues are Florida, Ohio, and Pennsylvania, which have in common rich caches of electoral votes and a coquettish reluctance to settle on one of their increasingly fervent suitors. Unsurprisingly, these states have been the three most frequent stops for both candidates. Conventional wisdom says Kerry can't win without Pennsylvania, which suggests he should concentrate all his energy there. But doing that would leave Florida and Ohio undefended and make it easier for Bush to win both. Maybe Kerry should foray into Ohio too, which might lead Bush to try to pick off Pennsylvania, which might divert his campaign's energy from Florida just enough for Kerry to snatch it away. ... You see the difficulty: As in any tactical problem, the best thing for Kerry to do depends on what Bush does, and the best thing for Bush to do depends on what Kerry does. At times like this, the division of mathematics that comes to our aid is game theory.

Game Theory and Politics (cont.)
To simplify our problem, let's suppose it's the weekend before Election Day and each candidate can only schedule one more visit. We'll concede Pennsylvania to Kerry; then for Bush to win the election, he must win both Florida and Ohio. Let's say that Bush has a 30 percent chance of winning Ohio and a 70 percent chance at Florida. Furthermore, we'll assume that Bush can increase his chances by 10 percent in either state by making a last-minute visit there, and that Kerry can do the same. If Bush and Kerry both visit the same state, then Bush's chances remain 30 percent in Ohio and 70 percent in Florida, and his chance of winning the election is 0.3 x 0.7, or 21 percent. If Bush visits Ohio and Kerry goes to Florida, Bush has a 40 percent chance in Ohio and a 60 percent chance in Florida, giving him a 0.4 x 0.6, or 24 percent chance of an overall win. Finally, if Bush visits Florida and Kerry visits Ohio, Bush's chances are 20 percent and 80 percent, and his chance of winning drops to 16 percent.

Example 3: Bush and Kerry
Kerry’s options Ohio Florida Bush’s Options 21% , 79% 24% , 76% 16% , 84% Bush’s dominant strategy is to visit Ohio. .3*.7 .4*.6 .2*.8 .3*.7 Nash Equilibrium: (Ohio, Ohio)

Real World More Complicated

EXAMPLE 4: Entry into a fast food market:
Is there a Nash Equilibrium(ia)? Yes, there are 2 – (Enter, Don’t Enter) and (Don’t Enter, Enter). Implies, no need for a dominant strategy to have NE. Burger King’s options Enter Skaneateles Don’t Enter Skaneateles McDonalds’ Options PBK = -40 PM = -30 PBK = 0 PM = 50 PBK = 40 PM = 0 NO Is there a dominant strategy for BK? NO Is there a dominant strategy for McD?

Example 5: Cournot Example from Last Class
Nash Equilibrium is Q1=26.67 and Q2=26.6 r1(Q2) Do Firms have a dominant Strategy? No, output that maximizes profits depends on output of other firm. 26.67 r2(Q1) 26.67

EXAMPLE 6: Monitoring Workers
Is there a Nash Equilibrium(ia)? Not a pure strategy Nash Equilibrium– player chooses to take one action with probability 1 Worker’s options Work Shirk Manager’s Options Monitor W: 1 M: -1 W: -1 M: 1 Don’t Monitor Randomize the actions yields a Nash = mixed strategy John Nash proved an equilibrium always exists NO Is there a dominant strategy for the worker? NO Is there a dominant strategy for the manager?

Mixed (randomized) Strategy
Definition: A strategy whereby a player randomizes over two or more available actions in order to keep rivals from being able to predict his or her actions.

Calculating Mixed Strategy EXAMPLE 6: Monitoring Workers
Manager randomizes (i.e. monitors with probability PM) in such a way to make the worker indifferent between working and shirking. Worker randomizes (i.e. works with probability Pw) in such a way as to make the manager indifferent between monitoring and not monitoring.

Example 6: Mixed Strategy
Worker’s options Work Shirk Manager’s Options Monitor W: 1 M: -1 W: -1 M: 1 Don’t Monitor 1-PW PW PM 1-PM

Manager selects PM to make Worker indifferent between working and shirking (i.e., same expected payoff) Worker’s expected payoff from working PM*(1)+(1- PM)*(-1) = -1+2*PM Worker’s expected payoff from shirking PM*(-1)+(1- PM)*(1) = 1-2*PM Worker’s expected payoff the same from working and shirking if PM=.5. This expected payoff is 0 (-1+2*.5=0 and 1-2*.5=0). Therefore, worker’s best response is to either work or shirk or randomize between working and shirking.

Manager’s expected payoff from monitoring PW*(-1)+(1- PW)*(1) = 1-2*PW
Worker selects PW to make Manager indifferent between monitoring and not monitoring. Manager’s expected payoff from monitoring PW*(-1)+(1- PW)*(1) = 1-2*PW Manager’s expected payoff from not monitoring PW*(1)+(1- PW)*(-1) = -1+2*PW Manager’s expected payoff the same from monitoring and not monitoring if PW=.5. Therefore, the manager’s best response is to either monitor or not monitor or randomize between monitoring or not monitoring .

Example 6A: What if costs of Monitoring decreases and Changes the Payoffs for Manager
Worker’s options Work Shirk Manager’s Options Monitor W: 1 M: -1 W: -1 M: 1 Don’t Monitor 1.5 -.5

Nash Equilibrium of Example 6A where cost of monitoring decreased
Worker works with probability and shirks with probability .375 (i.e., PW=.625) Same as in Ex. 5, Manager monitors with probability .5 and doesn’t monitor with probability .5 (i.e., PM=.5) The decrease in monitoring costs does not change the probability that the manager monitors. However, it increases the probability that the worker works.

Nash Equilibrium of Example 6
Worker works with probability .5 and shirks with probability .5 (i.e., PW=.5) Manager monitors with probability .5 and doesn’t monitor with probability .5 (i.e., PM=.5) Neither the Worker nor the Manager can increase their expected payoff by playing some other strategy (expected payoff for both is zero). They are both playing a best response to the other player’s strategy.

Example 7: Mixed Strategy and Tennis http://www. fuzzyyellowballs
Game: Server’s Possible Strategies: (Serve Left , Serve Right) Receiver’s Possible Strategies: (Defend Left , Defend Right) Receiver has a high probability of winning the point if she defends the side the server serves to.

Example 7: Mixed Strategy and Tennis (Payoffs are probability of winning point)
Receiver Defend Left Defend Right Server Serve Left .6 , .4 .75 , .25 Serve Right .7 , .3 .55 , .45 DL 1-DL SL 1-SL Mixed Strategy Equilibrium (SL =.5 , DL =.67) This results in the probability of the server winning the point to be .65 irrespective of whether he serves to the left or right.

Example 7: Mixed Strategy and Tennis What about the Real World?
Minimax Play at Wimbleton Walker and Wooders (AER 2001) “We use data from classic professional tennis matches to provide an empirical test of the theory of mixed strategy equilibrium. We find that the serve-and-return play of John McEnroe, Bjorn Borg, Boris Becker, Pete Sampras and others is consistent with equilibrium play.” Results: Probability Server wins is the same whether serve right or left. Which side server serves is not “serially independent”.

Example 8 A Beautiful Mind

Example 8: A Beautiful Mind
Other Student’s Options Pursue Blond Brunnette 1 Brunnette 2 John Nash’s 0 , 0 100 , 50 50 , 100 50 , 50 Nash Equilibria: (Pursue Blond, Pursue Brunnette 1) (Pursue Blond, Pursue Brunnette 2) (Pursue Brunnette 1, Pursue Blond) (Pursue Brunnette 2, Pursue Blond)

Sequential/Multi-Stage Games
Extensive form game: A representation of a game that summarizes the players, the information available to them at each stage, the strategies available to them, the sequence of moves, and the payoffs resulting from alternative strategies. (Often used to depict games with sequential play.)

Example 9 Potential Entrant Don’t Enter Enter Incumbent Firm
Potential Entrant: Incumbent: Price War Share Market (Hard) (Soft) Potential Entrant: Incumbent:

Example 9: With “Downstream” Actions
Potential Entrant Don’t Enter Enter Incumbent Firm Potential Entrant PIM1 PE1 Incumbent Firm Incumbent Firm PID1 PIM2 Potential Entrant and so on…. PE2 The present discounted value of profits for the incumbent and potential entrant depends on their strategies. Suppose each period the incumbent sets the optimal price as a monopolist and maximizes the present discounted value of profits which is +10. and so on….

What are the Nash Equilibria?
Potential Entrant Example 9 Don’t Enter Enter Incumbent Firm Potential Entrant: Incumbent: Price War Share Market (Hard) (Soft) Potential Entrant: Incumbent: What are the Nash Equilibria?

Nash Equilibria (Potential Entrant Enter,
Incumbent Firm Shares Market) (Potential Entrant Don’t Enter, Incumbent Firm Price War) Is one of the Nash Equilibrium more likely to occur? Why? Perhaps (Enter, Share Market) because it doesn’t rely on a non-credible threat.

Subgame Perfect Equilibrium
A condition describing a set of strategies that constitutes a Nash Equilibrium and allows no player to improve his own payoff at any stage of the game by changing strategies. (Basically eliminates all Nash Equilibria that rely on a non-credible threat – like Don’t Enter, Price War in Prior Game)

Example 9 Potential Entrant Don’t Enter Enter Incumbent Firm
Potential Entrant: Incumbent: Price War Share Market (Hard) (Soft) Potential Entrant: Incumbent: What is the Subgame Perfect Equilibrium? (Enter, Share Market)

Enter Don’t Enter Don’t
Big Ten Burrito Example 10 Enter Don’t Enter Chipotle Chipotle Enter Don’t Enter Don’t Enter Enter BTB: Chip:

Enter Don’t Enter Don’t
Big Ten Burrito Enter Don’t Enter Chipotle Chipotle Enter Don’t Enter Don’t Enter Enter BTB: Chip: Use Backward Induction to Determine Subgame Perfect Equilibrium.

Subgame Perfect Equilibrium
Chipotle should choose Don’t Enter if BTB chooses Enter and Chipotle should choose Enter if BTB chooses Don’t Enter. BTB should choose Enter given Chipotle’s strategy above. Subgame Perfect Equilibrium: (BTB chooses Enter, Chipotle chooses Don’t Enter if BTB chooses Enter and Enter if BTB chooses Don’t Enter.)

Example 11: Limit Pricing
When a firm sets it price and output so that there is not enough demand left for another firm to enter the market profitably.

Example 11: Lower Price, PL Monopoly Price, PM
Incumbent (suppose monopolist) Example 11: Lower Price, PL Monopoly Price, PM Potential Potential Entrant Entrant Don’t Don’t Enter Enter Enter Enter Incumbent Incumbent Hard Soft PL PM Hard Soft PL PM Ball Ball Ball Ball PE: Inc: Note: Incumbent’s profits are $10 per period if set monopoly price and $8 per period if set lower price. What price the incumbent sets initially does not influence second period profits for incumbent or potential entrant. For simplicity, second period payoffs are not discounted.

Example 11a: Lower Price, PL Monopoly Price, PM
Incumbent (suppose monopolist) Example 11a: Lower Price, PL Monopoly Price, PM Potential Potential Entrant Entrant Don’t Don’t Enter Enter Enter Enter Incumbent Incumbent Hard Soft PL PM Hard Soft PL PM Ball Ball Ball Ball PE: Inc: Note: Incumbent’s profits are $10 per period if set monopoly price and $8 per period if set lower price. What price the incumbent sets initially does not influence second period profits for incumbent or potential entrant. For simplicity, second period payoffs are not discounted.

Questions: Can you think of examples where the price the incumbent sets the first period could influence second period profits of the incumbent and perhaps the entrant? Are there other actions the incumbent can take prior to the potential entrant’s entry decision that could influence this decision? (R&D, Capital Investment, Lobbying, etc.)

Predatory Pricing Definition: When a firm first lowers its price in order to drive rivals out of business (and scare off potential entrants), and then raises its price when its rivals exit the market. What insights does the analysis on limit pricing provide for the logic of predatory pricing?

Slide from Oligopoly Lecture
Example 12 Slide from Oligopoly Lecture Firm 1’s Profits = 60*20-20*20=800 Firm 2’s Profits = 60*20-20*20=800 =AVC=ATC If firms collude on Q1=20 and Q2=20

Slide from Oligopoly Lecture
Example 12 Slide from Oligopoly Lecture Firm 1’s Profits = 50*30-20*30=900 Firm 2’s Profits = 50*20-20*20=600 =AVC=ATC Firms colluding is unlikely if they interact once because firms have incentive to cheat – in above case Firm 1 increases profits by cheating and producing 30 units.

Slide From Oligopoly Lecture
Repeated Interaction Suppose Firm 1 thinks Firm 2 won’t deviate from Q2=20 if Firm 1 doesn’t deviate from collusive agreement of Q1=20 and Q2=20. In addition, Firm 1 thinks Firm 2 will produce at an output of 80 in all future periods if Firm 1 deviates from collusive agreement of Q1=20 and Q2=20. Firm 1’s profits from not cheating Firm 1’s profits from cheating (by producing Q1=30 Today) … Today In 1 Year In 2 Years In 3 Years In 4 Years 800 … Today In 1 Year In 2 Years In 3 Years In 4 Years 900 Does Firm 2’s Strategy Rely on a Non-credible Threat? Depends on Game –unlikely to be credible even if infinitely repeated game

Use Backward Induction!!
What if Firms interact for 2 periods as Cournot Competitors? What is Subgame Perfect Equilibrium? Use Backward Induction!! In the second period, what will happen?

Cournot Equilibrium: Q1=26.67 and Q2=26.67
IN 2ND PERIOD!!!! r1(Q2) 26.67 r2(Q1) 26.67

Profits from Cournot Equilibrium: Q1=26.67 and Q2=26.67 so Q=Q1+Q2=53.3
Firm 1 Profits=46.66* *26.67= 713 Firm 2 Profits=46.66* *26.67= 713 46.66 =AVC=ATC 53.33

In the 1st period, what will happen?
If both firms realize that each will produce an output of in the 2nd period (resulting in profits of $713 for each firm) no matter what occurs in the 1st period, then the equilibrium the 1st period should be for both firms to produce and obtain profits of $713 the 1st period. Using this logic, the Subgame Perfect Equilibrium is for each firm to produce units of output the 1st period and units of output the 2nd period.

What if Firms interact for 1000 periods as Cournot Competitors
What if Firms interact for 1000 periods as Cournot Competitors? What is Subgame Perfect Equilibrium? Using similar logic as when the firms interact 2 periods, the Subgame Perfect Equilibrium is for each firm to produce units of output each period.

Do you really expect this type of outcome if the firms interact 1000 periods?
Laboratory experiments suggest that when facing a player a finite number of times, the players will “collude” for a number of periods. Many of these experiments involve a prisoners dilemma game being played a finite number of times.

In the real world, how do firms (and individuals) and individuals address the finite period problem?
Attempt to build in uncertainty associated with when the final period occurs. Attempt to “change game”.

Using Game Theory to Devise Strategies in Oligopolies that Increase Profits
Examples: Price Matching- advertise a price and promise to match any lower price offered by a competitor. Bertrand Oligopoly In the end, you would expect both firms to set a price of $20 (equal to MC) and have zero profits.

Using Game Theory to Devise Strategies in Oligopolies that Increase Profits
Examples: Price Matching- advertise a price and promise to match an lower price offered by a competitor. In Bertrand example, perhaps each firm would set a price of $60 and say will match. Induce Brand Loyalty – frequent flyer program Randomized pricing – inhibits consumers learning as to who offers lower price and reduces ability of competitors to undercut price.

Example 13: The Hold-Up Problem
Dan Conlin Invest in Firm Don’t Invest Specific Knowledge Dan Conlin and M&M negotiate salary Dan Conlin and M&M negotiate salary Dan Conlin: wI-CI wDI Marsh&McClennan: wI wDI Let wI and wDI denote Dan’s wage if he invests and doesn’t invest in the firm specific knowledge, respectively. Let the cost of investing for Dan be CI and let CI=30. Dan Conlin is worth 200 to M&M if he invests and is worth 150 if he doesn’t.

Dan Conlin Invest in Firm Don’t Invest Specific Knowledge Dan Conlin and M&M negotiate salary Dan Conlin and M&M negotiate salary Dan Conlin: wI-CI wDI Marsh&McClennan: wI wDI Assume that Dan’s best “outside option” is a wage of 100 whether or not he invests in the firm specific knowledge and that the outcome of the negotiations are such that Dan and M&M split the surplus. This means that wI=150 and wDI=125.

Dan Conlin Invest in Firm Don’t Invest Specific Knowledge Dan Conlin and M&M negotiate salary Dan Conlin and M&M negotiate salary Dan Conlin: wI-CI= wDI=125 Marsh&McClennan: wI = wDI= Subgame Perfect Equilibrium outcome has Dan Conlin not investing in the firm specific knowledge and receiving a wage of 125 even though the cost of the knowledge is 30 and it increases his value to the firm by 50.

Dan Conlin Invest in Firm Don’t Invest Specific Knowledge Dan Conlin and M&M negotiate salary Dan Conlin and M&M negotiate salary Dan Conlin: wI-CI= wDI=125 Marsh&McClennan: wI = wDI= What would you expect to happen in this case? Dan Conlin and M&M would divide cost of obtaining the knowledge.

Example 14: General Knowledge Investment
Dan Conlin Invest in Don’t Invest General Knowledge Dan Conlin and M&M negotiate salary Dan Conlin and M&M negotiate salary Dan Conlin: wI-CI= wDI =125 Marsh&McClennan: wI = wDI= Assume the game is as in the “hold-up” problem but that Dan’s best “outside option” is a wage of 120 if he invests in general knowledge and 100 if he does not. This means that wI=160 and wDI=125 (assuming split surplus when negotiate).

Example 14: General Knowledge Investment
Dan Conlin Invest in Don’t Invest General Knowledge Dan Conlin and M&M negotiate salary Dan Conlin and M&M negotiate salary Dan Conlin: wI-CI= wDI =125 Marsh&McClennan: wI = wDI= Subgame Perfect Equilibrium outcome has Dan Conlin investing in the general knowledge and receiving a wage of 160.

Example 15: Hold-up Problem (same idea as the Fisher Auto-body / GM situation)
Suppose there are two players: a computer chip maker (MIPS) and a computer manufacturer (Silicon Graphics). Initially, MIPS decides whether or not to customize its chip (the quantity of which is normalized to one) for a specific manufacturing purpose of Silicon Graphics. The customization costs $75 to MIPS, but adds value of $100 to the chip only when it is used by Silicon Graphics . The value of customization is partially lost when the chip is sold to an alternative buyer, who is willing to pay $60. If MIPS decides not to customize the chip, it can sell a standardized chip to Silicon Graphics at a price of zero and Silicon Graphics earns a payoff of zero from using the chip. If MIPS customizes the chip, the two players enter into a bargaining game where Silicon Graphics makes a take-it-or-leave-it price offer to MIPS. In response to this, MIPS can either accept the offer (in which case the game ends) or reject it (in which case MIPS approaches an alternative buyer who pays $60).

Example 15: Hold-Up Problem
MIPS Customize Don’t Customize Silicone Graphics 0 : MIPS 0 : Silicon Graphics Offer Price p MIPS Accept Reject MIPS: p = -15 Silicon Graphics: p Subgame Perfect Equilibrium – MIPS accepts price p if p>60. Silicone Graphics offers a price p=60. MIPS does not customize. The outcome of this game is that MIPS does not customize even though there is a surplus of $25 to be gained.

Is the Hold-Up Problem Applicable to other Situations?
YES Upstream Firm Investing in Specific Capital to produce input for Downstream Firm. Coal Mines located next to Power Plants. An academic buying a house before getting tenure or a big promotion. Taxing of Oil and Gas Lines by local jurisdictions. Multinational firms operating in foreign countries (Foreign Direct Investment) East Lansing Public Schools allocating a certain amount of money for capital expenditures and a certain amount for operating expenditures

Example 16: Tender Offer (obtained from John Morgan’s class website - “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.

Example 16: 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.

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

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

Example 17: 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.

Second-Price Sealed Bid Auction
Example 17: 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

TEAM 1 TEAM 2 Example 17: Ebay Game http://www.random.org/ Rd
Valuation Bid Payoff 1 2 3 4 5 6 AVE. Rd Valuation Bid Payoff 1 2 3 4 5 6 AVE

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

What if Bid, B, which is more than V? For Example, B=110 Payoff V=100 V=100 B=110 Highest Rival Bid

What if Bid, B, which is equal to V? Say V=100 so B=100 Payoff V=100 V=B=100 Highest Rival Bid

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.

(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.

2nd 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.

Sotheby’s Auctioning off Naismith Rules of Basketball http://www

Example 18: 1st 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.

Example 18: 1st Price Sealed Bid Auction Game
TEAM 1 TEAM 2 Rd Valuation Bid Payoff 1 2 3 4 5 6 AVE. Rd Valuation Bid Payoff 1 2 3 4 5 6 AVE

How Would Things Change if there were Three Bidders?
In 2nd 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 2nd price sealed bid auction).

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.

Example 19: Right of First Refusal (obtained from John Morgan’s class website - Many contracts contain a Right of First Refusal clause – including the collective bargaining agreement in the NBA pertaining to restricted free agents. 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).

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."

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 (OI1) which the player can accept or initially reject. If the player rejects OI1, the rival team decides whether to make an offer (OR) 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, OI2). 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.

OI1 OR OI2 I I, incumbent firm P, player R, rival firm Payoffs
RIGHT OF FIRST REFUSAL I I, incumbent firm P, player R, rival firm Payoffs (incumbent, player, rival) OI1 Initially Reject OI1 Accept OI1 P No Offer Make Offer R (10-OI1,OI1,0) Reject OI1 P R Accept OI1 OR I (10-OI1,OI1,0) (0,0,0) OI2 Accept OR Accept OI2 P Reject (10-OI2,OI2,-.5) (0,0,-.5) (0,OR,10-OR-.5)

Subgame Perfect – Backward Induction
Player: Accept OR if OR >O12 and OR >0 Accept Ol2 if O12 >ORand Ol2 >0 Reject if OR <0 and Ol2<0 Incumbent: Offer Ol2 =OR if OR<10 Offer Ol2 <OR if OR>10 Rival Firm: If make offer, offer OR<10 Rival Firm: No Offer Player (if rival firm makes no offer): Accept Ol1 if Ol1>0 Player: Accept Ol1 if Ol1 >0 Reject if Ol1 <0 Incumbent: Offer Ol1 =0 Outcome – Incumbent offers 0 and Player Accepts

What if there was not a Right of First Refusal?
Suppose the game is the same as before but now, if the player rejects OI1, and the rival firm makes an offer OR, the incumbent firm cannot make another offer. What is the likely outcome to this game?

OI1 OR I Payoffs (incumbent, player, rival) I, incumbent firm
NO RIGHT OF FIRST REFUSAL I Payoffs (incumbent, player, rival) I, incumbent firm P, player R, rival firm OI1 Initially Reject OI1 Accept OI1 P No Offer Make Offer R (10-OI1,OI1,0) Reject OI1 P R Accept OI1 OR Accept OR P (10-OI1,OI1,0) Accept OI1 (0,0,0) Reject (0,OR,10-OR-.5) (0,0,-.5) (10-OI1,OI1,-.5)

Subgame Perfect – Backward Induction
Player: Accept OR if OR >O11 and OR >0 Accept Ol1 if Ol1 >ORand Ol1 >0 Reject if OR <0 and Ol1<0 Rival Firm: Offer OR =Ol1 if Ol1 <9.5 Offer OR <Ol1 if Ol1 >9.5 Rival Firm: Make Offer if Ol1 <9.5 No Offer if Ol1 >9.5 Player (if rival firm makes no offer): Accept Ol1 if Ol1>0 Player: Accept Ol1 if Ol1 >9.5 Reject Ol1 if Ol1 <9.5 Incumbent: Offer Ol1 =9.5 Outcome – Incumbent offers 9.5 and Player Accepts

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

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? 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.

Game Theory.

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Game Theory.

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