The Bargain Whoever offers to another a bargain of any kind, proposes to do this. Give me that which I want, and you shall have this which you want …; and it is this manner that we obtain from one another the far greater part of those good offices we stand in need of. It is not from the benevolence of the butcher, the brewer, or the baker that we expect our dinner, but from their regard to their own interest. -- A. Smith, 1776

The Bargain Buyer and seller try to agree on a price. Buyer is better off at a price less than b, Seller at a price above s. If b > s, we say there is a positive zone of agreement, or Surplus: S(urplus) = b(uyer’s reservation price) – s(eller’s reservation price) b s i) b - s > 0 Surplus How to divide?

The Bargain Buyer and seller try to agree on a price. Buyer is better off at a price less than b, Seller at a price above s. If b > s, we say there is a positive zone of agreement, or Surplus: S(urplus) = b(uyer’s reservation price) – s(eller’s reservation price) b s i) b - s > 0 Surplus How to divide?

The Bargain Buyer and seller try to agree on a price. Buyer is better off at a price less than b, Seller at a price above s. If b > s, we say there is a positive zone of agreement, or Surplus: S(urplus) = b(uyer’s reservation price) – s(eller’s reservation price) b s i) b > s If b and s are known to both players: How should the surplus be divided?

The Bargain Buyer and seller try to agree on a price. Buyer is better off at a price less than b, Seller at a price above s. If b > s, we say there is a positive zone of agreement, or Surplus: S(urplus) = b(uyer’s reservation price) – s(eller’s reservation price) b s i) b > s If b and s are known to both players: How should the surplus be divided? Surplus = 50

The Bargain Buyer and seller try to agree on a price. Buyer is better off at a price less than b, Seller at a price above s. If b = s, we say the price is fully determined, and there is no room for negotiation. S(urplus) = b(uyer’s reservation price) – s(eller’s reservation price) b s ii) b = s

The Bargain Buyer and seller try to agree on a price. Buyer is better off at a price less than b, Seller at a price above s. If b < s, there is nothing to gained from the exchange. S(urplus) = b(uyer’s reservation price) – s(eller’s reservation price) b s (iii) b < sNo “zone of agreement”

The Bargain Buyer and seller try to agree on a price. Buyer is better off at a price less than b, Seller at a price above s. If b < s, there is nothing to gained from the exchange. S(urplus) = b(uyer’s reservation price) – s(eller’s reservation price) b s (iii) b < sNo “zone of agreement” What happens if information is incomplete?

We Play a Game PROPOSER RESPONDERPlayer # ____ Offer $ _____ Accept Reject

We Play a Game PROPOSER RESPONDERPlayer # ____ Offer $ _____ Accept Reject

We Play a Game PROPOSER RESPONDERPlayer # ____ Offer $ _____ Accept Reject

The Ultimatum Game OFFERS REJECTED ACCEPTED N = 20 Mean = $ Offers > 0 Rejected 1 Offer < 1.00 (20%) Accepted (3/6/00)

The Ultimatum Game OFFERS REJECTED ACCEPTED N = 33 Mean = $ Offers > 0 Rejected 1 Offer < $1 (20%) Accepted (2/28/01)

The Ultimatum Game OFFERS REJECTED ACCEPTED N = 37 Mean = $ Offers > 0 Rejected* 3 Offers < $1 (20%) Accepted (2/27/02) * 1 subject offered 0

The Ultimatum Game OFFERS REJECTED ACCEPTED N = 12 Mean = $ Offers > 0 Rejected 0 Offers < 1.00 (20%) Accepted (7/10/03)

The Ultimatum Game OFFERS REJECTED ACCEPTED N = 17 Mean = $ Offers > 0 Rejected 0 Offers < 1.00 (20%) Accepted (3/10/04)

The Ultimatum Game OFFERS REJECTED ACCEPTED N = 119 Mean = $ Offers > 0 Rejected 5/25 Offers < 1.00 (20%) Accepted Pooled data

The Ultimatum Game OFFERS REJECTED ACCEPTED N = 119 Mean = $ Offers > 0 Rejected 5/25 Offers < 1.00 (20%) Accepted Pooled data

The Ultimatum Game P 1 P What is the lowest acceptable offer? 8/8 4/4 21/23 2/2 3/3 18/26 13/15 N = 119 Mean = $ Offers > 0 Rejected 5/25 Offers < 1.00 (20%) Accepted Pooled data 5/6 3/17

The Ultimatum Game P 1 P What is the lowest acceptable offer? 8/8 4/4 21/23 2/2 3/3 18/26 13/15 N = 119 Mean = $ Offers > 0 Rejected 5/25 Offers < 1.00 (20%) Accepted Pooled data 5/6 3/17

The Ultimatum Game Theory predicts very low offers will be made and accepted. Experiments show: Mean offers are 30-40% of the total Mode = 50% Offers <20% are rare and usually rejected Guth Schmittberger, and Schwarze (1982) Kahnemann, Knetsch, and Thaler (1986) Also, Camerer and Thaler (1995)

The Ultimatum Game Theory predicts very low offers will be made and accepted. Experiments show: Mean offers are 30-40% of the total Mode = 50% Offers <20% are rare and usually rejected Guth Schmittberger, and Schwarze (1982) Kahnemann, Knetsch, and Thaler (1986) Also, Camerer and Thaler (1995) How would you advise Proposer?

The Ultimatum Game Theory predicts very low offers will be made and accepted. Experiments show: Mean offers are 30-40% of the total Mode = 50% Offers <20% are rare and usually rejected Guth Schmittberger, and Schwarze (1982) Kahnemann, Knetsch, and Thaler (1986) Also, Camerer and Thaler (1995) How would you advise Proposer? What do you think would happen if the game were repeated?

The Ultimatum Game How can we explain the divergence between predicted and observed results? Stakes are too low Fairness –Relative shares matter –Endowments matter –Culture, norms, or “manners” People make mistakes Time/Impatience

The Ultimatum Game How can we explain the divergence between predicted and observed results? Stakes are too low Fairness –Relative shares matter –Endowments matter –Culture, norms, or “manners” People make mistakes Time/Impatience

The Ultimatum Game How can we explain the divergence between predicted and observed results? Stakes are too low Fairness –Relative shares matter –Endowments matter –Culture, norms, or “manners” People make mistakes Time/Impatience

The Ultimatum Game How can we explain the divergence between predicted and observed results? Stakes are too low Fairness –Relative shares matter –Endowments matter –Culture, norms, or “manners” People make mistakes Time/Impatience

The Ultimatum Game How can we explain the divergence between predicted and observed results? Stakes are too low Fairness –Relative shares matter –Endowments matter –Culture, norms, or “manners” People make mistakes Time/Impatience

The Ultimatum Game How can we explain the divergence between predicted and observed results? Stakes are too low Fairness –Relative shares matter –Endowments matter –Culture, norms, or “manners” People make mistakes Time/Impatience

We Play Some Games PROPOSER RESPONDERPlayer # ____ Offer 2 or 5 Accept Reject (Keep 8 5)

We Play Some Games An offer to give 2 and keep 8 is accepted: PROPOSER RESPONDERPlayer # ____ Offer 2 or 5 Accept Reject (Keep 8 5)

Fair Play GAME AGAME B

Fair Play GAME CGAME D

Fair Play GAME AGAME B

Fair Play GAME CGAME D

Fair Play AB C D 50% /7 1/4 2/4 0/9 Rejection Rates, (8,2) Offer (5,5) (2,8) (8,2) (10,0) Alternative Offer 4/18/01, in Class. 24 (8,2) Offers 2 (5,5) Offers N = 26

Fair Play AB C D 50% /7 2/3 1/2 2/12 Rejection Rates, (8,2) Offer (5,5) (2,8) (8,2) (10,0) Alternative Offer 4/15/02, in Class. 24 (8,2) Offers 6 (5,5) Offers N = 30

Fair Play AB C D 50% Source: Falk, Fehr & Fischbacher, 1999 Rejection Rates, (8,2) Offer (5,5) (2,8) (8,2) (10,0) Alternative Offer

Fair Play What determines a fair offer? Relative shares Intentions Endowments Reference groups Norms, “manners,” or history

Fair Play These results show that identical offers in an ultimatum game generate systematically different rejection rates, depending on the other offer available to Proposer (but not made). This may reflect considerations of fairness: i) not only own payoffs, but also relative payoffs matter; ii) intentions matter. (FFF, 1999, p. 1 )

Bargaining Games Bargaining involves (at least) 2 players who face the the opportunity of a profitable joint venture, provided they can agree in advance on a division between them. Bargaining involves a combination of common as well as conflicting interests. The central issue in all bargaining games is credibility: the strategic use of threats, bluffs, and promises.

Bargaining Games Bargaining involves (at least) 2 players who face the the opportunity of a profitable joint venture, provided they can agree in advance on a division between them. Bargaining involves a combination of common as well as conflicting interests. The central issue in all bargaining games is credibility: the strategic use of threats, bluffs, and promises.

Bargaining Games Bargaining involves (at least) 2 players who face the the opportunity of a profitable joint venture, provided they can agree in advance on a division between them. Bargaining involves a combination of common as well as conflicting interests. The central issue in all bargaining games is credibility: the strategic use of threats, bluffs, and promises.

Bargaining Games P P 1 Disagreement point Two players have the opportunity to share $1, if they can agree on a division beforehand. Each writes down a number. If they add to $1, each gets her number; if not; they each get 0. Find the NE of this game. Divide a Dollar P 1 = x; P 2 = 1-x.

Bargaining Games P P 1 Disagreement point Two players have the opportunity to share $1, if they can agree on a division beforehand. Each writes down a number. If they add to $1, each gets her number; if not; they each get 0. Every division s.t. x + (1-x) = 1 is a NE. Divide a Dollar P 1 = x; P 2 = 1-x.

Subgame: a part (or subset) of an extensive game, starting at a singleton node (not the initial node) and continuing to payoffs. Subgame Perfect Nash Equilibrium (SPNE): a NE achieved by strategies that also constitute NE in each subgame. eliminates NE in which the players threats are not credible. selects the outcome that would be arrived at via backwards induction. Subgame Perfection

(0,0) (3,1) 1 2 Subgame Perfection (2,2) Chain Store Game A firm (Player 1) is considering whether to enter the market of a monopolist (Player 2). Player 2 can then choose to fight the entrant, or not. Enter Don’t Enter Fight Don’t Fight

(0,0) (3,1) 1 2 Subgame Perfection (2,2) Chain Store Game A firm (Player 1) is considering whether to enter the market of a monopolist (Player 2). Player 2 can then choose to fight the entrant, or not. Enter Don’t Enter Fight Don’t Fight Subgame

(0,0) (3,1) 1 2 Subgame Perfection (2,2) Chain Store Game Enter Don’t Fight Don’t 0, 0 3, 1 2, 2 2, 2 Fight Don’t Enter Don’t NE = {(E,D), (D,F)}.

(0,0) (3,1) 1 2 Subgame Perfection (2,2) Chain Store Game Enter Don’t Fight Don’t 0, 0 3, 1 2, 2 2, 2 Fight Don’t Enter Don’t NE = {(E,D), (D,F)}, but Fight for Player 2 is an incredible threat.

(0,0) (3,1) 1 2 Subgame Perfection (2,2) Chain Store Game Enter Don’t Fight Don’t 0, 0 3, 1 2, 2 2, 2 Fight Don’t Enter Don’t NE = {(E,D), (D,F)}.SPNE = {(E,D)}. Subgame Perfect Nash Equilibrium

A (ccept) 2 H (igh) 1 L (ow) R (eject) 5,5 0,0 8,2 0,0 Proposer (Player 1) can make High Offer (50-50%) or Low Offer (80-20%). Subgame Perfection Mini-Ultimatum Game

A (ccept) 2 H (igh) 1 L (ow) R (eject) H 5,5 0,0 5,5 0,0 L 8,2 0,0 0,0 8,2 5,5 0,0 8,2 0,0 AARRARRA Subgame Perfection Mini-Ultimatum Game

A (ccept) 2 H (igh) 1 L (ow) R (eject) H 5,5 0,0 5,5 0,0 L 8,2 0,0 0,0 8,2 5,5 0,0 8,2 0,0 AARRARRA Subgame Perfection Mini-Ultimatum Game

A (ccept) 2 H (igh) 1 L (ow) R (eject) H 5,5 0,0 5,5 0,0 L 8,2 0,0 0,0 8,2 5,5 0,0 8,2 0,0 AARRARRA Subgame Perfection Mini-Ultimatum Game

A (ccept) 2 H (igh) 1 L (ow) R (eject) H 5,5 0,0 5,5 0,0 L 8,2 0,0 0,0 8,2 5,5 0,0 8,2 0,0 AARRARRA Subgame Perfection Mini-Ultimatum Game

A (ccept) 2 H (igh) 1 L (ow) R (eject) H 5,5 0,0 5,5 0,0 L 8,2 0,0 0,0 8,2 5,5 0,0 8,2 0,0 AARRARRA Subgame Perfection Mini-Ultimatum Game

A (ccept) 2 H (igh) 1 L (ow) R (eject) H 5,5 0,0 5,5 0,0 L 8,2 0,0 0,0 8,2 AARRARRA 5,5 0,0 8,2 0,0 Subgame Perfect Nash Equilibrium SPNE = {(L,AA)} (H,AR) and (L,RA) involve incredible threats. Subgame Perfection Mini-Ultimatum Game

2 H 1 L 2 H 5,5 0,0 5,5 0,0 L 8,2 1,9 1,9 8,2 5,5 0,0 8,2 1,9 AARRARRA Subgame Perfection

2 H 1 L H 5,5 0,0 5,5 0,0 L 8,2 1,9 1,9 8,2 5,5 0,0 1,9 SPNE = {(H,AR)} AARRARRA Subgame Perfection

Alternating Offer Bargaining Game Two players are to divide a sum of money (S) is a finite number (N) of alternating offers. Player 1 (‘Buyer’) goes first; Player 2 (‘Seller’) can either accept or counter offer, and so on. The game continues until an offer is accepted or N is reached. If no offer is accepted, the players each get zero. A. Rubinstein, 1982

Alternating Offer Bargaining Game Two players are to divide a sum of money (S) is a finite number (N) of alternating offers. Player 1 (‘Buyer’) goes first; Player 2 (‘Seller’) can either accept or counter offer, and so on. The game continues until an offer is accepted or N is reached. If no offer is accepted, the players each get zero. 1 (a,S-a) 2 (b,S-b) 1 (c,S-c) (0,0)

Alternating Offer Bargaining Game 1 (a,S-a) 2 (b,S-b) 1 (c,S-c) (0,0) S = $5.00 N = 3

Alternating Offer Bargaining Game 1 (a,S-a) 2 (b,S-b) 1 (4.99, 0.01) (0,0) S = $5.00 N = 3

Alternating Offer Bargaining Game 1 (4.99,0.01) 2 (b,S-b) 1 (4.99,0.01) (0,0) S = $5.00 N = 3 SPNE = (4.99,0.01) The game reduces to an Ultimatum Game

Now consider what happens if the sum to be divided decreases with each round of the game (e.g., transaction costs, risk aversion, impatience). Let S = Sum of money to be divided N = Number of rounds  = Discount parameter Shrinking Pie Game

S = $5.00 N = 3  = (a,S-a) 2 (b,S-b) 1 (c,S-c) (0,0)

Shrinking Pie Game S = $5.00 N = 3  = (3.74,1.26)2 (1.25, 1.25) 1 (1.24,0.01) (0,0) 1

Shrinking Pie Game S = $5.00 N = 3  = (3.74,1.26)2 (1.25, 1.25) 1 (1.24,0.01) (0,0) 1

Shrinking Pie Game S = $5.00 N = 3  = (3.74,1.26)2 (1.25, 1.25) 1 (1.24,0.01) (0,0) 1

Shrinking Pie Game S = $5.00 N = 3  = (3.74,1.26)2 (1.25, 1.25) 1 (1.24,0.01) (0,0) 1

Shrinking Pie Game S = $5.00 N = 4  = (3.13,1.87)2 (0.64,1.86) 1 (0.63,0.62) 2 (0.01, 0.61) (0,0) 1

Shrinking Pie Game P 1 P N = 1 (4.99, 0.01) 2(2.50, 2.50) 3(3.74, 1.26) 4(3.12, 1.88) 5(3.43, 1.57)… This series converges to (S/(1+  ), S – S/(1+  )) = (3.33, 1.67) This pair {S/(1+  ),S-S/(1+  )} are the payoffs of the unique SPNE. for  = ½ 1

Shrinking Pie Game P 1 P N = 1 (4.99, 0.01) 2(2.50, 2.50) 3(3.74, 1.26) 4(3.12, 1.88) 5(3.43, 1.57)… This series converges to (S/(1+  ), S – S/(1+  )) = (3.33, 1.67) This pair {S/(1+  ),S-S/(1+  )} are the payoffs of the unique SPNE. for  = ½ 1 2

Shrinking Pie Game P 1 P N = 1 (4.99, 0.01) 2(2.50, 2.50) 3(3.74, 1.26) 4(3.12, 1.88) 5(3.43, 1.57)… This series converges to (S/(1+  ), S – S/(1+  )) = (3.33, 1.67) This pair {S/(1+  ),S-S/(1+  )} are the payoffs of the unique SPNE. for  = ½ 1 2 3

Shrinking Pie Game P 1 P N = 1 (4.99, 0.01) 2(2.50, 2.50) 3(3.74, 1.26) 4(3.13, 1.87) 5(3.43, 1.57)… This series converges to (S/(1+  ), S – S/(1+  )) = (3.33, 1.67) This pair {S/(1+  ),S-S/(1+  )} are the payoffs of the unique SPNE. for  = ½

Shrinking Pie Game P 1 P N = 1 (4.99, 0.01) 2(2.50, 2.50) 3(3.74, 1.26) 4(3.13, 1.87) 5(3.43, 1.57)… This series converges to (S/(1+  ), S – S/(1+  )) = (3.33, 1.67) This pair {S/(1+  ),S-S/(1+  )} are the payoffs of the unique SPNE. for  = ½

Shrinking Pie Game P 1 P N = 1 (4.99, 0.01) 2(2.50, 2.50) 3(3.74, 1.26) 4(3.13, 1.87) 5(3.43, 1.57)… This series converges to (S/(1+  ), S – S/(1+  )) = (3.33, 1.67) This pair {S/(1+  ),S-S/(1+  )} are the payoffs of the unique SPNE. for  = ½

Shrinking Pie Game Optimal Offer (O*) expressed as a share of the total sum to be divided = [S-S/(1+  )]/S O* =  /(1+  SPNE = {1- [  /(1+  )],  /(1+  )} Thus both  =1 and  =0 are special cases of Rubinstein’s model: When  =1 (no bargaining costs), O* = 1/2 When  =0, game collapses to the ultimatum version and O* = 0 (+  )

Shrinking Pie Game

Bargaining Games Bargaining games are fundamental to understanding the price determination mechanism in “small” markets. The central issue in all bargaining games is credibility: the strategic use of threats, bluffs, and promises. When information is asymmetric, profitable exchanges may be “left on the table.” In such cases, there is an incentive to make oneself credible (e.g., appraisals; audits; “reputable” agents; brand names; lemons laws; “corporate governance”).

Acquiring a Company BUYER represents Company A (the Acquirer), which is currently considering make a tender offer to acquire Company T (the Target) from SELLER. BUYER and SELLER are going to be meeting to negotiate a price. Company T is privately held, so its true value is known only to SELLER. Whatever the value, Company T is worth 50% more in the hands of the acquiring company, due to improved management and corporate synergies. BUYER only knows that its value is somewhere between 0 and 100 ($/share), with all values equally likely. Source: M. Bazerman

Acquiring a Company BUYER represents Company A (the Acquirer), which is currently considering make a tender offer to acquire Company T (the Target) from SELLER. BUYER and SELLER are going to be meeting to negotiate a price. Company T is privately held, so its true value is known only to SELLER. Whatever the value, Company T is worth 50% more in the hands of the acquiring company, due to improved management and corporate synergies. BUYER only knows that its value is somewhere between 0 and 100 ($/share), with all values equally likely. Source: M. Bazerman

Acquiring a Company What offer should Buyer make?

Acquiring a Company 5 Source:Bazerman, BU MBA Students $ Offers

Acquiring a Company $ Source:Bazerman, BU MBA Students Similar results from MIT Master’s Candidates CPA; CEOs. Offers

Acquiring a Company There are many ways to depict the decision tree for this game. Buyer moves first and can make any Offer from  to . Let’s say Buyer offers $60. O(ffer) = Buyer Seller Accept Reject O – s = EP(O) = - 15 The expected payoff of a $60 offer is a net loss of - $15.

Acquiring a Company There are many ways to depict the decision tree for this game. Buyer moves first and can make any Offer from  to . Seller accepts if O > s. O(ffer) = Buyer Seller Accept Reject Chance s 60 s = 0 60 O – s = EP(O) = - 15 The expected payoff of a $60 offer is a net loss of - $15.

Acquiring a Company There are many ways to depict the decision tree for this game. Buyer moves first and can make any Offer from  to . Let’s say Buyer offers $60. O(ffer) = Buyer Seller Accept Reject Chance s 60 s = 0 60 s – O = EP(O) = - 15 The expected payoff of a $60 offer is a net loss of - $15.

Acquiring a Company OFFER VALUE ACCEPT OR VALUE GAIN OR TO SELLER REJECTTO BUYER LOSS (O) (s) (3/2 s = b) (b - O) $60 $0 A $0 $ A A A A A R R - -

Acquiring a Company The key to the problem is the asymmetric information structure of the game. SELLER knows the true value of the company (s). BUYER knows only the upper and lower limits (0 < s < 100). Therefore, buyer must form an expectation on s (s'). BUYER also knows that the company is worth 50% more under the new management, i.e., b' = 3/2 s'. BUYER makes an offer (O). The expected payoff of the offer, EP(O), is the difference between the offer and the expected value of the company in the hands of BUYER: EP(O) = b‘ – O = 3/2s‘ – O.

Acquiring a Company BUYER wants to maximize her payoff by offering the smallest amount (O) she expects will be accepted: EP(O) = b‘ – O = 3/2s‘ – O. O = s' + .Seller accepts if O > s. Now consider this: Buyer has formed her expectation based on very little information. If Buyer offers O and Seller accepts, this considerably increases Buyer’s information, so she can now update her expectation on s. How should Buyer update her expectation, conditioned on the new information that s < O?

Acquiring a Company BUYER wants to maximize her payoff by offering the smallest amount (O) she expects will be accepted: EP(O) = b‘ – O = 3/2s‘ – O. O = s' + .Seller accepts if O > s. Now consider this: Buyer has formed her expectation based on very little information. If Buyer offers O and Seller accepts, this considerably increases Buyer’s information, so she can now update her expectation on s. How should Buyer update her expectation, conditioned on the new information that s < O?

Acquiring a Company BUYER wants to maximize her payoff by offering the smallest amount (O) she expects will be accepted: EP(O) = b‘ – O = 3/2s‘ – O. O = s' + .Seller accepts if O > s. Let’s say BUYER offers $50. If SELLER accepts, BUYER knows that s cannot be greater than (or equal to) 50, that is: 0 < s < 50. Since all values are equally likely, s''/(s < O) = 25. The expected value of the company to BUYER (b'' = 3/2s'' = 37.50), which is less than the 50 she just offered to pay. (EP(O) = ) When SELLER accepts, BUYER gets a sinking feeling in the pit of her stomach.

Acquiring a Company BUYER wants to maximize her payoff by offering the smallest amount (O) she expects will be accepted: EP(O) = b‘ – O = 3/2s‘ – O. O = s' + .Seller accepts if O > s. Let’s say BUYER offers $50. If SELLER accepts, BUYER knows that s cannot be greater than (or equal to) 50, that is: 0 < s < 50. Since all values are equally likely, s''/(s < O) = 25. The expected value of the company to BUYER (b'' = 3/2s'' = 37.50), which is less than the 50 she just offered to pay. (EP(O) = ) When SELLER accepts, BUYER gets a sinking feeling in the pit of her stomach. THE WINNER’S CURSE!

Acquiring a Company BUYER wants to maximize her payoff by offering the smallest amount (O) she expects will be accepted: EP(O) = b‘ – O = 3/2s‘ – O. O = s' + .Seller accepts if O > s. Generally: EP(O) = O - ¼s' (-  ). EP is negative for all values of O. THE WINNER’S CURSE!

Acquiring a Company The high level of uncertainty swamps the potential gains available, such that value is often left on the table, i.e., on average the outcome is inefficient. Under these particular conditions, BUYER should not make an offer. SELLER has an incentive to reveal some information to BUYER, because if BUYER can reduce the uncertainty, she may make an offer that leaves both players better off.

Bargaining Games In real-world negotiations, players often have incomplete, asymmetric, or private information, e.g., only the seller of a used car knows its true quality and hence its true value. Making agreements is made all the more difficult “when trust and good faith are lacking and there is no legal recourse for breach of contract” (Schelling, 1960: 20). Rubinstein’s solution: If a bargaining game is played in a series of alternating offers, and if a speedy resolution is preferred to one that takes longer, then there is only one offer that a rational player should make, and the only rational thing for the opponent to do is accept it immediately!