Chapter 7 Bargaining “Necessity never made a good bargain”
Economic Markets • Allocation of scarce resources • Many buyers & many sellers traditional markets • Many buyers & one seller auctions • One buyer & one seller bargaining
The Move to Game-Theoretic Bargaining • Baseball • Each side submits an offer to an arbitrator who must chose one of the proposed results • Meet-in-the-Middle • Each side proposes its “worst acceptable offer” and a deal is struck in the middle, if possible • Forced Final • If an agreement is not reached by some deadline, one party makes a final take-it-or-leave-it offer
Bargaining & Game Theory • Art: Negotiation • Science: Bargaining • Game theory’s contribution: • to the rules of the encounter
Outline • Importance of rules: The rules of the game determine the outcome • Diminishing pies: The importance of patience • Estimating payoffs: Trust your intuition
Take-it-or-leave-it Offers • Consider the following bargaining game (over a cake): • I name a take-it-or-leave-it split. • If you accept, we trade • If you reject, no one eats! • Faculty senate – if we can’t agree on a recommendation (for premiums in health care) the administration will say, “There is no consensus” and do what they want. We will have no vote. • Under perfect information, there is a simple rollback equilibrium
Take-it-or-leave-it Offers • Second period: Accept if p > 0 • First period: Offer smallest possible p The “offerer” keeps all profits 1-p , p accept p reject 0 , 0
Counteroffers and Diminishing Pies • In general, bargaining takes on a “take-it-or-counteroffer” procedure • If time has value, both parties prefer to trade earlier to trade later • E.g. Labor negotiations – Later agreements come at a price of strikes, work stoppages, etc. • Delays imply less surplus left to be shared among the parties
Two Stage Bargaining • Bargaining over division of a cake • I offer a proportion, p, of the cake to you • If rejected, you may counteroffer (and of the cake melts) • Payoffs: • In first period: 1-p , p • In second period: (1-)(1-p) , (1-)p
Rollback • Since period 2 is the final period, this is just like a take-it-or-leave-it offer: • You will offer me the smallest piece that I will accept, leaving you with all of 1- and leaving me with almost 0 • What do I do in the first period?
Rollback • Give you at least as much surplus • Your surplus if you accept in the first period is p • Accept if: Your surplus in first period Your surplus in second period p 1-
Rollback • If there is a second stage, you get 1- and I get 0. • You will reject any offer in the first stage that does not offer you at least 1-. • In the first period, I offer you 1-. • Note: the more patient you are (the slower the cake melts) the more you receive now!
First or Second Mover Advantage? • Are you better off being the first to make an offer, or the second?
Example: Cold Day • If =1/5 (20% melts) • Period 2: You offer a division of 1,0 • You get all of remaining cake = 0.8 • I get 0 = 0 • In the first period, I offer 80% • You get 80% of whole cake = 0.8 • I get 20% of whole cake = 0.2
Example: Hot Day • If =4/5 (80% melts) • Period 2: You offer a division of 1,0 • You get all of remaining cake = 0.2 • I get 0 = 0 • In the first period, I offer 20% • You get 20% of whole cake = 0.2 • I get 80% of whole cake = 0.8
First or Second Mover Advantage? • When players are impatient (hot day) First mover is better off • Rejecting my offer is less credible since we both lose a lot • When players are patient (cold day) Second mover better off • Low cost to rejecting first offer • Either way – if both players think through it, deal struck in period 1
Don’t Waste Cake • Why doesn’t this happen? • Reputation building • Lack of information COMMANDMENT In any bargaining setting, strike a deal as early as possible!
Uncertainty in Civil Trials • Civil Lawsuits • If both parties can predict the future jury award, can settle for same outcome and save litigation fees and time • If both parties are sufficiently optimistic, they do not envision gains from trade Plaintiff sues defendant for $1M • Legal fees cost each side $100,000 • If each agrees that the chance of the plaintiff winning is ½: • Plaintiff: $500K - $100K = $ 400K • Defendant: - $500K - $100K = $-600K • If simply agree on the expected winnings, $500K, each is better off settling out of court. • Defendant should just give the plaintiff $400K as he saves $200K.
Uncertainty in Civil Trials • What if both parties are too optimistic? • Each thinks that his or her side has a ¾ chance of winning: • Plaintiff: $750K - $100K = $ 650K • Defendant: - $250K - $100K = $-350K • No way to agree on a settlement! Defendant would be willing to give plaintiff $350, but plaintiff won’t accept.
von Neumann/Morganstern Utility over wealth • How big is the cake? • Is something really better than nothing?
Lessons • Rules of the bargaining game uniquely determine the bargaining outcome • Which rules are better for you depends on patience, information • What is the smallest acceptable piece? Trust your intuition • Delays are always less profitable: Someone must be wrong
Non-monetary Utility • Each side has a reservation price • Like in civil suit: expectation of winning • The reservation price is unknown • One must: • Consider non-monetary payoffs • Probabilistically determine best offer • But – probability implies a chance that no bargain will be made
Example: Uncertain Company Value • Company annual profits are either $150K or $200K per employee • Two types of bargaining: • Union makes a take-it-or-leave-it offer • Union makes an offer today. If it is rejected, the Union strikes, then makes another offer • A strike costs the company 10% of annual profits
Take-it-or-leave-it Offer • Probability that the company is “highly profitable,” i.e. $200K is p • If offer wage of $150 • Definitely accepted • Expected wage = $150K • If offer wage of $200K • Accepted with probability p • Expected wage = $200K(p)
Take-it-or-leave-it OfferExample I • p=9/10 • 90% chance company is highly profitable • Best offer: Ask for $200K wage • Expected value of offer: (.9)$200K = $180K • But: 10% chance of No Deal!
Take-it-or-leave-it OfferExample II • p=1/10 • 10% chance company is highly profitable • Best offer: Ask for $150K wage • If ask for $200K Expected value of offer: (.1)$200K = $20K • If ask for $150K, get $150K • Not worth the risk to ask for more.
Two-period Bargaining • If first-period offer is rejected: A strike costs the company 10% of annual profits • Note: strike costs a high-value company more than a low-value company! • Use this fact to screen!
Screening in Bargaining • What if the Union asks for $160K in the first period? • Low-profit firm ($150K) rejects – as can’t afford to take. • High-profit firm must guess what will happen if it rejects: • Best case – Union strikes and then asks for only $140K (willing to pay for some cost of strike, but not all) • In the mean time – Strike cost the company $20K • High-profit firm accepts
Separating Equilibrium • Only high-profit firms accept in the first period • If offer is rejected, Union knows that it is facing a low-profit firm • Ask for $140K in second period • Expected Wage: • $170K (p) + $140K (1-p) • In order for this to be profitable • $170K (p) + $140K (1-p) > 150K • 140 +(170-140)p = 140+ 30p >150 • if p > 1/3 , you win
What’s Happening • Union lowers price after a rejection • Looks like “Giving in” • Looks like Bargaining • Actually, the Union is screening its bargaining partner • Different “types” of firms have different values for the future • Use these different values to screen • Time is used as a screening device
Bargaining • The non cooperative games miss something essential: people can make deals - then can agree to behave in a way that is better for both. Economics is based on the fact that there are many opportunities to "gain from trade“. • With the opportunities, however comes the possibility of being exploited. Human beings have developed a systems of contracts and agreements, as well as institutions that enforce those agreements. • Cooperative game theory is about games with enforceable contracts.
Strategic Decisions • Non-strategic decisions are those in which one’s choice set is defined irrespective of other people’s choices. • Strategic decisions are those in which the choice set that one faces and/or the outcomes of such choices depend on what other people do. These decisions can be characterised in two general ways: • Cooperative games: Where the outcome is agreed upon through joint action and enforced by some outside arbitrator. • Non-cooperative games: The outcome arises through separate action, and thus does not rely on outside arbitration.
Cooperative Bargaining • A bargaining situation can be approached as a cooperative game. All bargaining situations have two things in common: • The total payoff created through cooperation must be greater than the sum of each party’s individual payoff that they could achieve separately. • The bargaining is thus over the ‘surplus’ payoff. As no bargaining party would agree to getting less than what they get on their own. A player’s ‘outside option’ is also known as a BATNA (Best Alternative To Negotiated Agreement) or disagreement value.
Two people dividing cash CONSIDER THE FOLLOWING BARGAINING GAME • Jenny and George have to divide candy bar • They have to agree how to divide up the candy • If they do not agree they each get nothing • They can’t divide up more than the whole thing • They could leave some candy on the table What is the range of likely bargaining outcomes?
Likely range of outcomes • Clearly neither Jenny nor George can individually get more than 100% • Further, neither of them can get less than zero – either could veto and avoid the loss • Finally, it would be silly to agree on something that does not divide up the whole 100% – they could both agree to something better • But that is about as far as our prediction can go!
Likely range of outcomes • So our prediction is that Jenny will get %j and George will get %g where • %j ≥ 0; • %g ≥ 0 and; • %j + %g = %100.
Modified bargaining game • Jenny and George still have to divide 100% • They must agree to any split • If they do not agree then Jenny gets nothing and George gets 50% • They can’t divide up more than 100% • They could leave some on the table • Now, what is the range of likely bargaining outcomes?
Likely range of outcomes in modified game • Clearly neither Jenny nor George can individually get more than 100% • Further, Jenny would veto anything where she gets less than 0% • George will veto anything where he gets less than 50% • And it would be silly to agree on something that does not divide up the whole 100%
Likely range of outcomes for modified game • So our prediction is that Jenny will get %j and George will get %g where • %j ≥ 0; • %g ≥ 50 and; • %j + %g = %100. • Note by changing George’s ‘next best alternative’ to agreeing with Jenny, we change the potential bargaining outcomes.
Ultimatum Games:Basic Experimental Results • In a review of numerous ultimatum experiments Camerer (2003) found: • The results reported…are very regular. Modal and median ultimatum offers are usually 40-50 percent and means are 30-40 percent. There are hardly any offers in the outlying categories of 0, 1-10%, and the hyper-fair category 51-100%. Offers of 40-50 percent are rarely rejected. Offers below 20 percent or so are rejected about half the time.”
Ultimatum Bargaining withIncomplete Information • Player 1 begins the game by drawing a chip from the bag. Inside the bag are 30 chips ranging in value from $1.00 to $30.00. Player 1 then makes an offer to Player 2. The offer can be any amount in the range from $0.00 up to the value of the chip. • Player 2 can either accept or reject the offer. If accepted,Player 1 pays Player 2 the amount of the offer and keeps the rest. If rejected, both players get nothing.
Experimental Results • Questions: • How much should Player 1 offer Player 2? • Does the amount of the offer depend on the size of the chip? • 2) What should Player 2 do? • Should Player 2 accept all offers or only offers above a specified amount? • Explain.
How should Ali & Baba split the pie? • Ali and Baba have to decide how to split up an ice cream pie. • The rules specify that Ali begins by making an offer on how to split the pie. Baba can then either accept or reject the offer. • If Baba accepts the offer, the pie is split as specified and the game is over. • If Baba rejects the offer, the pie shrinks, since it is ice cream, and Baba must then make an offer to Ali on how to split the pie. • Ali can either accept or reject this offer. • If rejected, the pie shrinks again and Ali must then make another offer to Baba. • This procedure is repeated until and offer is accepted or the pie is gone.
How should Ali & Baba split the pie? • 1. How much should Ali offer Baba in the first round? • 2. Should Baba accept this offer? Why or why not?
Ali & Baba’s Pie Woes • Initial Pie Size = 100 • Pie decreases by 20 each time an offer is rejected. • Question: What is the optimal split of this pie? That is, how much should Ali offer Baba in the first round so that Baba will accept the offer.
Ali & Baba’s Pie Woes Baba may as well accept first offer. It never really gets better for him.
Formulas: If the number of rounds in the game is even, the pie should be split 50/50. If the number of rounds in the game is odd, then the proportion of the pie for each player is: (n + 1)/2n for Ali (initial offer) – first person advantage! (n-1)/2n for Baba. For example, in this game n = 5, so Ali gets: (5+1) / (2*5) = 6/10. 60% of 100 is 60.
Suppose the discount is 25% If Ali offered 50%, Baba would have no reason to question! He never gets more.
Model for Bargaining – no shrinking pieExample – two people bargaining over goods • Amy has 10 apples and 2 banana • Betty has 1 apple and 15 bananas • Before eating their fruit, they meet together Questions: • Can Amy and Betty agree to exchange some fruit? • If so, how do we characterize the likely set of possible trades between Amy and Betty?