Lecture Notes 19: Bargaining
Basic Principles
The basis of any bargaining problem is a surplus. The surplus is the total amount by which the players’ total payoffs are higher by making a successful bargain. The surplus should be positive. In other words, for bargaining to make sense, players must be doing better than they would be without making a deal.
Bargaining is not a zero-sum game. It’s true that, conditional on an agreement being made, bargaining is a zero-sum game over how to split the surplus. But making an agreement is better for everyone than not making an agreement because it allows the full surplus to be realized.
As a motivating example, suppose that Firm A produces a processor that it can sell on its own for $900. Firm B produces a software package that it can sell on its own for $100. But Firm A and Firm B together can produce a computer that it could sell for $3000. The surplus from this deal is $2000. This is the amount by which total profits are higher by the two firms making a deal with each other. Any deal has to give Firm A at least $900 and Firm B at least $100, but how to split up the other $2000 is not clear. This is the subject of bargaining problems.
Just a bit of history on this problem. When “many” are matched against “one”, economists have generally understood that the concentrated side of the market will be able to extract all of the surplus from the deal. The obvious examples are monopoly and monopsony. Buyers have no bargaining power individually against a huge monopoly. But “one” against “one” was always viewed as a difficult problem because there is no obvious rule for splitting up the surplus.
Nash Bargaining
Two players, A and B, get together to split a total value of 𝑉𝑉. If A and B cannot reach a deal with each other, then A’s best outside option is worth 𝑎𝑎 and B’s best outside option is worth 𝑏𝑏. The values 𝑎𝑎 and 𝑏𝑏 represent each player’s BATNA (best alternative to negotiated agreement). This is the best that players can do if no deal is reached. For a concrete example, imagine that 𝑉𝑉 is the total value of a marriage, while 𝑎𝑎 and 𝑏𝑏 represent the utility of each person from remaining single.
In the Nash Bargaining model, each player gets his BATNA plus some share of the surplus. Let ℎ and 𝑘𝑘 designate the fraction of the surplus claimed by player A and player B, respectively. By definition, ℎ + 𝑘𝑘 = 1.
Letting 𝑋𝑋 indicate the final allocation to player A and letting 𝑌𝑌 indicate the final allocation to player B, we can write the bargaining allocations as follows. Remember, each player has to get his BATNA plus some share of the surplus.
𝑋𝑋 = 𝑎𝑎 + ℎ(𝑉𝑉 − 𝑎𝑎 − 𝑏𝑏) 𝑌𝑌 = 𝑏𝑏 + 𝑘𝑘(𝑉𝑉 − 𝑎𝑎 − 𝑏𝑏)
If we rearrange slightly and divide, we can obtain:
𝑋𝑋 − 𝑎𝑎 𝑌𝑌 − 𝑏𝑏 =
ℎ 𝑘𝑘
In other words, the surplus to each agent (the allocation over and above the BATNA) is split in the ratio ℎ: 𝑘𝑘.
The backstop is the outcome without any negotiation. In other words, the backstop is the BATNA payoff vector (𝑋𝑋, 𝑌𝑌) = (𝑎𝑎, 𝑏𝑏). This is an inefficient outcome in the sense that the total value realized is lower than if a bargain is reached, which is the whole basis of trying to reach a bargain. The efficient frontier is not attained unless a bargain is reached.
Along the efficient frontier, the outcome of the final bargain will depend on the bargaining shares
ℎ and 𝑘𝑘. The easiest way to see the solution graphically is to draw a ray through the backstop point with slope 𝑘𝑘
ℎ. The final outcome will be where that ray intersects the efficient frontier. For example:
• If ℎ = 0, then player A has no bargaining power. In this case, the 𝑘𝑘
ℎ line is vertical. In the final outcome, player A gets only his BATNA 𝑎𝑎, with the entire surplus going to player B.
• If 𝑘𝑘 = 0, then player B has no bargaining power. In this case, the 𝑘𝑘
ℎ line is horizontal. In the final outcome, player B gets only his BATNA 𝑏𝑏, with all the surplus going to player A.
Variable-Threat Bargaining
There are various ways to get a better outcome for yourself in a bargaining game. One way is to
change the bargaining shares. In terms of the graphical representation, this will tilt the 𝑘𝑘
ℎ line in your favor.
A second way is for you to increase your BATNA. Increasing your BATNA improves your final allocation even if the bargaining shares stay the same, by giving you a better outside option. This point is illustrated below, where player A’s BATNA increases from 𝑎𝑎 to 𝑎𝑎′.
This point is easy to understand intuitively. Suppose that you are worth $60,000 to your employer and that your next-best job offer is $40,000. If the surplus is split evenly, then your employer will offer you $50,000. On the other hand, if you have an outside offer for $55,000, then the surplus from the deal is reduced to $5000. If the surplus is split in half, your employer will offer you $57,500 to take the job. Even with the same 50/50 split in the bargaining shares, you are better off when your outside option is stronger.
A third option is to lower the other player’s BATNA. For example, if you make yourself so important to a company that it will go bankrupt without you, then the backstop point (the outcome if they don’t make a deal with you) is very bad for the company. This helps you in the bargaining process.
Alternating-Offer Models: Example 1
The Nash bargaining model lays out the foundations for describing bargaining problems. But, in a sense, the Nash Bargaining model is more of a description of the problem than a solution because it does not give us any hint about what the bargaining shares ℎ and 𝑘𝑘 should be. That’s the difficult part of the problem. When a surplus exists, how can we rationalize the division of the surplus?
One common class of models is known as alternating-offer models. There are two important features of these models.
• Bargaining goes back and forth between the two parties – one party makes an offer, the other party responds, etc… Practically, this is how most bargaining proceeds.
• There is a decay in the total surplus available the longer the bargaining takes. Again, this comports with reality in many ways. Imagine time costs or litigation costs associated with delays. The longer labor contract negotiations take, for example, the more legal fees are incurred and longer the business cannot operate, which reduces the total profit available once a bargain is reached. It is also basic economics that future values are worth less than present values.
Here is a simple example to start with. Suppose that two parties are bargaining over how to split a surplus of 10. They alternate making offers for the division, with player 1 moving first. However, for each round the deal is delayed, the total surplus available is reduced by 1. Thus, if 10 rounds pass and there is still no deal, then there is no surplus left and the game is over. The table below summarizes how negotiations proceed in this game.
Round Offer By Surplus
1 P1 10
2 P2 9
3 P1 8
4 P2 7
5 P1 6
6 P2 5
7 P1 4
8 P2 3
9 P1 2
10 P2 1
• In round 10, player 2 offers the split (0,1). Player 1 accepts since he receives nothing anyway if he does not accept.
One important note. Here and throughout, we assume that players accept offers when they are indifferent. If you don’t like this, imagine player 2 offering player 1 some infinitesimal amount. The point is that player 1 has no credible commitment to reject any offer since he gets 0 by rejecting. Knowing this, player 2 should make the offer for player 1 infinitesimally small.
• In round 9, player 1 must offer player 2 at least 1 since player 2 can guarantee herself a payoff of 1 by holding out until the next round. Player 1 keeps the rest for himself and offers the split (1,1), which player 2 accepts.
• In round 8, player 2 must offer player 1 at least 1 since this is what he will get by rejecting and holding out until the next round. Player 2 keeps the rest for herself and offers the split
(1,2), which player 1 accepts.
• In round 7, player 1 must offer player 2 at least 2 since this is what she will get by rejecting and holding out until the next round. Player 1 keeps the rest for himself and offers the split
(2,2), which player 2 accepts.
• In round 6, player 2 must offer player 1 at least 2 since this is what he will get by rejecting and holding out until the next round. Player 2 keeps the rest for herself and offers the split
(2,3), which player 1 accepts.
• In round 5, player 1 must offer player 2 at least 3 since this is what she will get by rejecting and holding out until the next round. Player 1 keeps the rest for himself and offers the split
(3,3), which player 2 accepts.
• In round 4, player 2 must offer player 1 at least 3 since this is what he will get by rejecting and holding out until the next round. Player 2 keeps the rest for herself and offers the split
(3,4), which player 1 accepts.
• In round 3, player 1 must offer player 2 at least 4 since this is what she will get by rejecting and holding out until the next round. Player 1 keeps the rest for himself and offers the split
(4,4), which player 2 accepts.
• In round 2, player 2 must offer player 1 at least 4 since this is what he will get by rejecting and holding out until the next round. Player 2 keeps the rest for herself and offers the split
• In round 1, player 1 must offer player 2 at least 5 since this is what she will get by rejecting and holding out until the next round. Player 1 keeps the rest for himself and offers the split
(5,5), which player 2 accepts.
Despite all this work we did, the outcome is just that player 1 offers an even split (5,5) in the first round and player 2 accepts immediately. We don’t actually reach any of the subsequent rounds. But, as always, in sequential games we have to solve from the end in order to figure out what happens at the beginning.
An important feature of alternating-offer models is that a bargain is reached immediately. Players realize that their prospects only get worse by delaying. Another important feature of these models is that they rationalize a half/half division of the surplus. Nash bargaining did not give us any insight about what the bargaining shares for dividing the surplus, ℎ and 𝑘𝑘, should be. We might have an instinctive preference for splitting the surplus in half, but the alternating offers model actually produces this as the equilibrium outcome.
If we consider a slight modification where the number of periods is odd, rather than even, we get almost the same answer. Consider the same alternating-offers setup, with a total surplus of 5 that decays by 1 each period.
Round Offer By Surplus
1 P1 5
2 P2 4
3 P1 3
4 P2 2
5 P1 1
• In round 5, player 1 offers (1,0). Player 2 accepts since she gets nothing anyway by rejecting.
• In round 4, player 2 has to give player 1 at least 1, since he can get this by rejecting and holding out until the next round. Player 2 keeps the rest for herself, offering (1,1), which player 1 accepts.
• In round 2, player 2 has to give player 1 at least 2, since he can get this by rejecting and holding out until the next round. Player 2 keeps the rest for herself, offering (2,2), which player 1 accepts.
• In round 1, player 1 has to give player 2 at least 2, since she can get this by rejecting and holding out until the next round. Player 1 keeps the rest for himself, offering (3,2), which player 2 accepts.
Again, the outcome is that player 1 initially offers the split (3,2), and player 2 accepts this offer immediately. Generally, the outcome in these arithmetic decay alternating-offers models is for the players either to split the surplus in half, or with a slight advantage to the first mover if the number of periods is odd rather than even.
Alternating-Offers Models: Example 2
Consider a hotel where management is negotiating with a labor union over how the profits will be divided. The season lasts 100 days and the hotel’s profit is $1000 for each day it is open. The hotel will open as soon as there is an agreement about how to split the profits. The union and the management alternate offers, with the union moving first. The table below summarizes.
Round Offer By Surplus
1 Union 100,000
2 Management 99,000
3 Union 98,000
4 Management 97,000
5 Union 96,000
⋮ ⋮ ⋮
96 Management 5000
97 Union 4000
98 Management 3000
99 Union 2000
100 Management 1000
We again apply backwards induction, starting from period 100.
• In period 99, the union has to offer management at least 1000 since management can get 1000 by holding out until the last period. The union keeps the rest, offering (1000,1000).
• In period 98, management has to offer the union at least 1000 since the union can get 1000 by holding out until period 99. Management keeps the rest, offering (1000,2000).
• In period 97, the union has to offer management at least 2000 since management can get 2000 by holding out until period 98. The union keeps the rest, offering (2000,2000).
• In period 96, management has to offer the union at least 2000 since the union can get 2000 by holding out until period 97. Management keeps the rest, offering (2000,3000).
Fill in the end of the table to see the pattern.
Round Offer By Surplus Offer
1 Union 100,000
2 Management 99,000
3 Union 98,000
4 Management 97,000
5 Union 96,000
⋮ ⋮ ⋮
96 Management 5000 (2000,3000)
97 Union 4000 (2000,2000)
98 Management 3000 (1000,2000)
99 Union 2000 (1000,1000)
100 Management 1000 (0,1000)
Round Offer By Surplus Offer
1 Union 100,000 (50000,50000)
2 Management 99,000 (49000,50000)
3 Union 98,000 (49000,49000)
4 Management 97,000 (48000,49000)
5 Union 96,000 (48000,48000)
⋮ ⋮ ⋮
96 Management 5000 (2000,3000)
97 Union 4000 (2000,2000)
98 Management 3000 (1000,2000)
99 Union 2000 (1000,1000)
100 Management 1000 (0,1000)
The ultimate outcome of the game is that the union offers on day 1 to split the profits with $50,000 each to the union and to the management, and management accepts this offer immediately.
Alternating-Offers Models: Example 3
Reconsider the same example as above, but suppose now that the union has an outside option. Workers can earn $300 a day in unemployment compensation for each day that they are off the job and a deal is not made. The alternation and available totals are the same as in the previous example. Again, we work backwards.
• On day 100, management must offer the union at least 300, because this is what they could get by holding out and getting unemployment pay. Management keeps the rest for itself, so its offer is (300,700).
• On day 99, the union must offer management at least 700, because this is what management could get by holding out until the last period. The union keeps the rest for itself, so the union’s offer is (1300,700).
• On day 97, the union must offer management at least 1400, because this is what management could get by holding out until day 98. The union keeps the rest for itself, so the union’s offer is (2600,1400).
• On day 96, management must offer the union at least 2900. This is to compensate them for the 2600 they could get by holding until day 97 out and for the 300 in foregone unemployment compensation that the union could collect by holding out. Management keeps the rest for itself, so management’s offer is (2900,2100).
Fill in the offers from the last few periods to see the pattern.
Round Offer By Surplus Offer
1 Union 100,000
2 Management 99,000
3 Union 98,000
4 Management 97,000
5 Union 96,000
⋮ ⋮ ⋮
96 Management 5000 (2900,2100)
97 Union 4000 (2600,1400)
98 Management 3000 (1600,1400)
99 Union 2000 (1300,700)
100 Management 1000 (300,700)
Round Offer By Surplus Offer
1 Union 100,000 (65000,35000)
2 Management 99,000 (64000,35000)
3 Union 98,000 (63700,34300)
4 Management 97,000 (62700,34300)
5 Union 96,000 (62400,33600)
⋮ ⋮ ⋮
96 Management 5000 (2900,2100)
97 Union 4000 (2600,1400)
98 Management 3000 (1600,1400)
99 Union 2000 (1300,700)
100 Management 1000 (300,700)
Thus, the outcome is that the union in round 1 proposes to split the profits with $65,000 going to the union and $35,000 going to the management, and management accepts this offer immediately.
In bargaining language, what is going on here is that the presence of unemployment pay for the union increases their BATNA. In other words, the union’s outside option is not quite as bad since they can get $300 per day in unemployment pay. The surplus is split half/half, but the daily surplus from a successful negotiation is only $700, not the full $1000. The union gets their $300 BATNA plus $350 of the surplus. Management gets its $0 BATNA plus $350 of the surplus. This is what leads to the final 65/35 split of the revenues. The revenues are not split in half, but the surplus is – only part of the total revenues are surplus since the union has a nonzero outside option.
Manipulating Information in Bargaining Games
We will close this section with some practical advice on playing bargaining games.
• If you have a high BATNA, you want to signal it right away. If you have a low BATNA, it’s better to hide it. For example, if you are selling your house and you already have a nice offer from one of your neighbors, you should tell potential buyers right away. But you probably wouldn’t want to tell them that you have no other offers.
• The only real test of the other side’s BATNA is acceptance of an offer. What if you make an offer on a house and the seller accepts it right away? Maybe you should have offered a lower amount! But you’d probably look like a jerk to renege on your initial offer.
• You might think about making an offer that is only good for a fixed period in order to make it difficult for the other party to seek outside options. But how credible is this? If you offer to buy a house for $200,000 today, will you really refuse to buy it next week for $200,000 just because the clock you set ran out? Pulling this off requires a credible commitment.
• You can signal your patience by being patient. If you give counteroffers right away, the other side might think you are eager to get the deal done and that you would accept a lower share of the surplus. Maybe it’s better for you to wait a little bit to respond so that the other side won’t think it’s urgent for you to make a deal right away.
• You can signal a high BATNA by threatening to walk away. But, again, how credible is a threat to walk away if you want the deal too?
• Claim that you are bound by a mandate, which reduces the room you have to negotiate. For example, a recruiter for a firm might claim that the company has a policy that prohibits him from making wage offers above a certain level. A husband looking at a house might say that that his wife won’t allow him to spend more than a certain amount.
• Think about using brinkmanship. Threatening to scuttle a bargain outright may not be credible, but the risk of a breakdown in negotiations may be.
One important point is that a lot of these strategies contain risk. If you delay offers and threaten to walk away when you are buying a house, the seller may find another buyer in the meantime. If you signal a BATNA that is too high, then the other person might think that there is no room for negotiating with you and might walk away completely.
Multi-Issue Bargaining
Theoretically, bringing more issues to the table in bargaining can increase the potential for mutually beneficial deals. For example, suppose that a buyer likes the furniture in a house very much but that the furniture has little value to the seller. Even if they can’t agree on a selling price for the house, throwing the furniture in might give them a chance to come to an agreement.
the US at the same time. Japan is very strong on trade. But military issues are a weak point for Japan that the US could potentially leverage when trying to extract trade concessions, so Japan will not negotiate the two together.
Multi-Party Bargaining
Multi-party bargaining can theoretically facilitate negotiations because parties can seek a circle of accommodations that create mutual benefits for everyone involved. For example, the US produces wheat but needs cars. Japan produces cars but needs oil. Saudi Arabia produces oil but needs wheat. No two of them together can work out a mutually beneficial exchange, but the three of them jointly can. The US can ship wheat to Saudi Arabia, which would ship oil to Japan, which in turn would ship cars to the US.
The difficulty with this arrangement is enforcement. If the US fails to offer good terms on its wheat for Saudi Arabia, then Saudi Arabia really has no recourse because it can only withhold something from Japan, not from the US.
Evidence on Bargaining
Games with gradual decay, like those we have considered in this section, lead to an approximately equal division of the surplus. But what about games with rapid decay, where the surplus falls quickly? The extreme example is an ultimatum game, which is basically a bargaining game with complete decay after one period. This creates a huge first-mover advantage for the player making the offer, who can keep everything for himself, which is the equilibrium outcome.
Laboratory evidence consistently contradicts the equilibrium result for ultimatum games. In defiance of the very clear SPE of the game, the most common outcome is a 50/50 split. Very few offers worse than 75/25 are ever made, and most of those offers are rejected.
Explaining the contradiction between the game’s equilibrium and actual laboratory play is a point of long-standing contention. Here are some of the possible interpretations.
• People aren’t smart enough to do backwards induction – Player 1 doesn’t realize that player 2 has no credible threat to reject offers.
• People get a self-esteem boost from rejecting low offers, and this should be included in the payoff numbers.
• Fairness and equality is a social norm, and there is a positive payoff from obeying social norms which should be reflected in the payoff numbers.
Lecture Notes 19 Appendix: Rubinstein Bargaining
The Rubinstein bargaining model is an alternating-offer bargaining model. The difference with our previous examples is that the decay is geometric rather than arithmetic. The discount factor is
𝛿𝛿. In other words, a payoff of 𝑉𝑉 that is received 𝑛𝑛 periods from today has a value of only 𝛿𝛿𝑛𝑛𝑉𝑉, where the discount factor is 0 < 𝛿𝛿 < 1. Future payoffs are worth less than current payoffs.
To keep the notation simple, we will normalize the size of the surplus available to be divided as 1. In other words, there is always a total of 1 available, but if a deal is not made immediately then its value decays over time due to discounting. By convention, the Rubinstein bargaining model starts in period 0. This is so that the current period is undiscounted, and then the discounting in any future period 𝑡𝑡 is given by 𝛿𝛿𝑡𝑡.
The table below shows the sequence of offers and the total payoff sum available. Again, the total surplus available is always equal to 1, but is discounted if it is not received until the future. The game lasts for 𝑇𝑇 periods. If no agreement is reached after 𝑇𝑇 periods, then the game is over and both players receive a payoff of 0. We will work out the case where 𝑇𝑇 is even.
Round Offer By Payoff Available
0 P1 1
1 P2 𝛿𝛿
2 P1 𝛿𝛿2
⋮ ⋮ ⋮
𝑇𝑇 − 4 P1 𝛿𝛿𝑇𝑇−4
𝑇𝑇 − 3 P2 𝛿𝛿𝑇𝑇−3
𝑇𝑇 − 2 P1 𝛿𝛿𝑇𝑇−2
𝑇𝑇 − 1 P2 𝛿𝛿𝑇𝑇−1
𝑇𝑇 P1 𝛿𝛿𝑇𝑇
We solve backwards to find the SPE.
• In period 𝑇𝑇 − 1, player 2 must give player 1 a payoff of at least 𝛿𝛿𝑇𝑇 since player 1 can guarantee this for himself by holding out. Since the total payoff available in period 𝑇𝑇 − 1 is 𝛿𝛿𝑇𝑇−1, this means that player 2 can keep 𝛿𝛿𝑇𝑇−1− 𝛿𝛿𝑇𝑇 for herself.
• In period 𝑇𝑇 − 2, player 1 must give player 2 a payoff of at least 𝛿𝛿𝑇𝑇−1− 𝛿𝛿𝑇𝑇 since player 2 can guarantee this for herself by holding out. The total payoff available is 𝛿𝛿𝑇𝑇−2, so player 1 can keep for himself 𝛿𝛿𝑇𝑇−2− (𝛿𝛿𝑇𝑇−1− 𝛿𝛿𝑇𝑇) = 𝛿𝛿𝑇𝑇−2− 𝛿𝛿𝑇𝑇−1+ 𝛿𝛿𝑇𝑇
• In period 𝑇𝑇 − 3, player 2 must give player 1 a payoff of at least 𝛿𝛿𝑇𝑇−2− 𝛿𝛿𝑇𝑇−1+ 𝛿𝛿𝑇𝑇 since player 1 can guarantee this for himself by holding out until the next period. Since the total payoff available is 𝛿𝛿𝑇𝑇−3, player 2 can keep for herself 𝛿𝛿𝑇𝑇−3− (𝛿𝛿𝑇𝑇−2− 𝛿𝛿𝑇𝑇−1+ 𝛿𝛿𝑇𝑇) =
𝛿𝛿𝑇𝑇−3− 𝛿𝛿𝑇𝑇−2+ 𝛿𝛿𝑇𝑇−1− 𝛿𝛿𝑇𝑇
• In period 𝑇𝑇 − 4, player 1 must give player 2 a payoff of at least 𝛿𝛿𝑇𝑇−3− 𝛿𝛿𝑇𝑇−2+ 𝛿𝛿𝑇𝑇−1− 𝛿𝛿𝑇𝑇 since player 2 can guarantee this for herself by holding out. Since the total payoff available is 𝛿𝛿𝑇𝑇−4, player 1 can keep for himself 𝛿𝛿𝑇𝑇−4− (𝛿𝛿𝑇𝑇−3− 𝛿𝛿𝑇𝑇−2+ 𝛿𝛿𝑇𝑇−1− 𝛿𝛿𝑇𝑇) = 𝛿𝛿𝑇𝑇−4−
𝛿𝛿𝑇𝑇−3+ 𝛿𝛿𝑇𝑇−2− 𝛿𝛿𝑇𝑇−1+ 𝛿𝛿𝑇𝑇
Let’s enter this information on a table to see the pattern.
Round Offer By Payoff for Offer
0 P1
1 P2
2 P1
⋮ ⋮ ⋮
𝑇𝑇 − 4 P1 (𝛿𝛿𝑇𝑇−4− 𝛿𝛿𝑇𝑇−3+ 𝛿𝛿𝑇𝑇−2− 𝛿𝛿𝑇𝑇−1+ 𝛿𝛿𝑇𝑇, 𝛿𝛿𝑇𝑇−3− 𝛿𝛿𝑇𝑇−2+ 𝛿𝛿𝑇𝑇−1− 𝛿𝛿𝑇𝑇) 𝑇𝑇 − 3 P2 (𝛿𝛿𝑇𝑇−2− 𝛿𝛿𝑇𝑇−1+ 𝛿𝛿𝑇𝑇, 𝛿𝛿𝑇𝑇−3− 𝛿𝛿𝑇𝑇−2+ 𝛿𝛿𝑇𝑇−1− 𝛿𝛿𝑇𝑇) 𝑇𝑇 − 2 P1 (𝛿𝛿𝑇𝑇−2− 𝛿𝛿𝑇𝑇−1+ 𝛿𝛿𝑇𝑇, 𝛿𝛿𝑇𝑇−1− 𝛿𝛿𝑇𝑇) 𝑇𝑇 − 1 P2 (𝛿𝛿𝑇𝑇, 𝛿𝛿𝑇𝑇−1− 𝛿𝛿𝑇𝑇)
𝑇𝑇 P1 (𝛿𝛿𝑇𝑇, 0)
We have a little bit of accounting left to do because this table shows the payoff value of the offers that are tendered in each successive period. But, because of discounting, the actual offer that is tendered is not the same as the payoff value of the offer.
For example, in the final period 𝑇𝑇, the offer tendered was (1,0), but this led to a payoff value of
Generally, in any period 𝑡𝑡, the relationship is that any offer tendered 𝑉𝑉 has a payoff value of 𝛿𝛿𝑡𝑡𝑉𝑉 in that period. Thus, to convert “backwards” from the payoff value to the actual offer, we need to divide the payoff expression by 𝛿𝛿𝑡𝑡.
• In period 𝑇𝑇 − 1, we need to divide through the payoffs by the relevant discount factor 𝛿𝛿𝑇𝑇−1 in order to convert from payoff value back to the actual offer.
(𝛿𝛿𝑇𝑇, 𝛿𝛿𝑇𝑇−1− 𝛿𝛿𝑇𝑇) → (𝛿𝛿, 1 − 𝛿𝛿)
• In period 𝑇𝑇 − 2, we need to divide through the payoffs by the relevant discount factor 𝛿𝛿𝑇𝑇−2 in order to convert from payoff value back to the actual offer.
(𝛿𝛿𝑇𝑇−2− 𝛿𝛿𝑇𝑇−1+ 𝛿𝛿𝑇𝑇, 𝛿𝛿𝑇𝑇−1− 𝛿𝛿𝑇𝑇) → (1 − 𝛿𝛿 + 𝛿𝛿2, 𝛿𝛿 − 𝛿𝛿2)
Follow this same procedure to convert the payoff value of the offers from each period back to the current value of the offer in that period.
Round Offer By Offer
0 P1
1 P2
2 P1
⋮ ⋮ ⋮
𝑇𝑇 − 4 P1 (1 − 𝛿𝛿 + 𝛿𝛿2− 𝛿𝛿3+ 𝛿𝛿4, 𝛿𝛿 − 𝛿𝛿2+ 𝛿𝛿3− 𝛿𝛿4) 𝑇𝑇 − 3 P2 (𝛿𝛿 − 𝛿𝛿2+ 𝛿𝛿3, 1 − 𝛿𝛿 + 𝛿𝛿2− 𝛿𝛿3) 𝑇𝑇 − 2 P1 (1 − 𝛿𝛿 + 𝛿𝛿2, 𝛿𝛿 − 𝛿𝛿2)
𝑇𝑇 − 1 P2 (𝛿𝛿, 1 − 𝛿𝛿)
𝑇𝑇 P1 (1,0)
Now it’s easy to see the pattern of alternating signs. Taking the pattern back to period 0, in which player 1 makes an offer, his initial offer is:
(1 − 𝛿𝛿 + 𝛿𝛿2− 𝛿𝛿3+ ⋯ + 𝛿𝛿𝑇𝑇, 𝛿𝛿 − 𝛿𝛿2+ 𝛿𝛿3+ ⋯ − 𝛿𝛿𝑇𝑇)
This is the division that is offered initially by player 1, and the division is accepted immediately by player 2.
�1 + 𝛿𝛿1 + 𝛿𝛿 ,𝑇𝑇+1 𝛿𝛿 − 𝛿𝛿1 + 𝛿𝛿 �𝑇𝑇+1
Suppose now that the bargaining can go on for a long time. As 𝑇𝑇 → ∞, the terms 𝛿𝛿𝑇𝑇+1 approach 0 since 0 < 𝛿𝛿 < 1. Thus, in the limit, as the bargaining process gets longer, the SPE offer tendered in the first period and accepted immediately by player 2 is:
�1 + 𝛿𝛿 ,1 1 + 𝛿𝛿�𝛿𝛿
There is an interesting property of this result. As 𝛿𝛿 → 1, the division of the surplus approaches
�12,12�. But as 𝛿𝛿 → 0, the division of the surplus approaches (1,0).
What is the interpretation? For values of 𝛿𝛿 close to 1, players are patient. Thus, player 1 has to offer a relatively even division of the surplus in order to get player 2 to agree to the division. This is because player 2 is patient and faces very little cost of holding out.
But for values of 𝛿𝛿 close to 0, players are impatient. Player 1 has a big first-mover advantage in this case. He realizes that holding out is extremely costly for player 2. Thus, he can offer a very uneven division initially (in his favor), since it is costly for player 2 to reject and delay the settlement. She will accept an uneven offer rather than delay. The extreme case of high discounting is an ultimatum game where the value goes to 0 if an agreement is not reached initially. This allows player 1 to capture all of the surplus since player 2 has no credible threat to reject any offers.
What if the players have different discount factors? Let 𝛿𝛿1 be player 1’s discount factor and let 𝛿𝛿2 be player 2’s discount factor. In the infinitely-repeated Rubinstein bargaining problem, backwards induction analysis similar to what we did above shows that the SPE is for the following offer to be tendered by player 1 in the first period and accepted immediately by player 2.
�1 − 𝛿𝛿1 − 𝛿𝛿2
1𝛿𝛿2,
𝛿𝛿2(1 − 𝛿𝛿1)
1 − 𝛿𝛿1𝛿𝛿2 �
Problems
1. John is a computer expert who is negotiating an employment contract. The contract specifies John’s salary 𝑤𝑤. John’s payoff is 𝑤𝑤. John produces $200,000 of output for the firm, so the firm’s payoff is 200,000 − 𝑤𝑤. If John and the firm fail to reach an agreement, the firm gets a payoff of 0 (there are no other employees with John’s skill set), but John has an outside salary offer of $60,000. John’s bargaining share is 𝜋𝜋𝐽𝐽 and the firm’s bargaining share is 𝜋𝜋𝐹𝐹, where 𝜋𝜋𝐽𝐽+ 𝜋𝜋𝐹𝐹 = 1.
a. What salary does the firm offer to John in the Nash bargaining solution? How does it depend on 𝜋𝜋𝐽𝐽?
b. What would happen to John’s salary offer if his outside offer were higher than $60,000? Explain using Nash bargaining reasoning.
c. What would happen to John’s salary offer if he became more productive to the firm? Explain using Nash bargaining reasoning.
2. Players 1 and 2 are bargaining over how to split a total surplus of 10. The available sum decays by 1 for each period in which a deal is not made. The players alternate offers, with the following twist: Player 1 makes offers in two successive periods, then player 2 makes one offer, then player 1 makes offers in the next two periods, then player 2 makes one offer, etc… In what period is a deal accepted, and what is the division of the surplus?
4. Consider again the bargaining situation from the notes of a hotel that is open for 100 days and that generates $1000 of profit for each day that it is open. The hotel cannot open until the union and the management make a deal about how to split the profits. We considered the case where neither party had an outside option, and then we considered the case where the workers had an outside option to collect $300 per day in unemployment compensation. Suppose for this problem that the hotel management also has an outside option. Workers can collect $300 per day in unemployment compensation without a deal, but management can also earn $500 in profit per day without a deal by using some non-union source of labor. In the SPE of this new game, on what day is a deal accepted and what is the division of the profits? Give a bargaining interpretation of the surplus division.
5. (The pirate problem) Five pirates – A, B, C, D and E – must decide how to distribute 100 gold coins that they just plundered. The pirates follow a strict hierarchy, with A being the most senior pirate and E being the least senior pirate. The rules are as follows. Pirate A proposes a distribution of the coins. If a majority vote for his distribution, the coins are disbursed and the game is over. If a majority vote against his distribution, then he is thrown overboard and Pirate B has the same opportunity to propose a distribution. The remaining pirates vote on this distribution and the problem continues as such, all the way to the last pirate. Any tie votes are resolved in favor of the proposer. Each pirate’s first goal is to survive, and beyond this each pirate’s goal is to acquire as many coins as possible. Also, pirates enjoy watching people being thrown overboard. Thus, unlike a usual bargaining problem, the pirates prefer to throw someone overboard if they are indifferent. Which pirate’s offer is accepted and what is the offer?
