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What price compromise?

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Abstract

This paper identifies, and tests experimentally, a prediction of the Nash bargaining axioms that may appear counterintuitive. The context is a simple bargaining problem in which two players have to agree a choice from three alternatives. One alternative favours one player and a second favours the other. The third is an apparently reasonable compromise, but is in fact precluded as an agreed choice by the Nash axioms. Experimental results show that agreement on this third alternative occurs rather often. Our subjects’ behaviour could be interpreted as the paying of an irrationally high price, according to the Nash axioms, in order to reach a compromise agreement.

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Notes

  1. This has to be distinguished from individual rationality. If agreement and thus the avoidance of \(z\) requires unanimous individual assent, then each partner assenting to C is equilibrium in individual strategies, even with five or more red balls.

  2. The obvious attraction of Bag C is its ex ante fairness; hence our prior conjecture. But our purpose here is not to elaborate or test theories of fairness. Simply, whatever the attraction of Bag C it is ineligible as an agreed choice according to the Nash axioms. See also footnote 5, below.

  3. Recent examples include Shimer (2005) and Atakan (2006).

  4. There are some important caveats to this assertion. Firstly, the Nash theory does not require that individual utilities are vNM, but only that they have the same degree of cardinality as vNM, as implied by the axiom of Invariance. In the present context, however, vNM is a natural interpretation. Secondly, Nash (1950) included an axiom of Symmetry, in effect implying additionally that \(\rho =1/2\); the non-symmetric generalisation was provided by Kalai (1977). Thirdly, conventional analysis would require X to have a compact, convex image in utility space, guaranteeing a unique argmax of (1). Zhou (1997) generalised to a class of non-convex cases, but not including our CCP. So the theoretical foundation of the appeal here to the Nash Product is not secure, which is why we describe it as only an indicative demonstration.

  5. It might be conjectured that this conclusion could be avoided by relaxing the assumption of self-interestedness. To the extent that the Nash theory is one of agreeable and/or (at least in its symmetric version) fair compromise between self-interested bargainers, this would be a questionable move. But in any case the conjecture is false. Assume a single draw, so that J winning and K winning are mutually exclusive. Assume also that J and K have identical vNM preferences such that the utility value of one’s self winning is 1, of the other person winning is \(\alpha \in [0,1]\), and of no-one winning is 0. Then, with the Nash Products correspondingly reformulated, \(\hbox {N}(\hbox {C})\ge \hbox {N}(\hbox {A})\) and \(\hbox {N}(\hbox {C})\ge \hbox {N}(\hbox {B})\) together imply:

    $$\begin{aligned} \gamma ^{2}\;\;\ge \;\;\frac{[\pi +\alpha (1 - \pi )][1 - \pi +\alpha \pi ]}{(1+\alpha )^{2}}\;\;=\;\;\frac{\alpha +(1-\alpha )^{2}\pi (1 - \pi )}{(1+\alpha )^{2}}. \end{aligned}$$

    The required minimum value of \(\gamma \) here increases with \(\alpha \). So in this sense other-regarding preferences make Bag C actually less, rather than more, agreeable. The intuition is perhaps clearest at \(\alpha =1\), where each player cares only that either, rather than neither, of them wins the prize. Thus, both J and K are indifferent between Bags A and B, each of which delivers the prize to one of them for sure, and they both prefer either of these to Bag C with its attendant possibility, however small, of no-one winning.

  6. For individual choice, \(\Gamma (\hbox {X},d)\) would be expected or required to be invariant to \(d\), the default outcome in the event of failure to choose.

  7. The correspondence between our axioms and the standard Nash axioms is blurred by specification of the latter being within the framework of a (unique) bargaining solution defined in utility space. Our alternative framework provides for a relatively simple and direct explanation of the ineligibility (given \(\gamma <\hat{{\pi }})\) of Bag C in the CCP.

  8. This axiom would follow from the Nash construction of a bargaining solution being defined in utility space, given that these are vNM utilities (see note 4 above).

  9. Camerer (1995) provides an excellent survey.

  10. Risk aversion cannot be an explanation for the choice of Bag C, neither can loss-aversion with a fixed reference point (for example, Bag C), neither can certain non-EU preferences such as fanning-out. But this leaves other possibilities, for example, fanning-in preferences or a shifting reference point. We leave it to others to investigate whether these are the explanations.

  11. Two key references are Roth and Malouf (1979) and Roth and Murnighan (1982). For full references and an overview, see pp. 40–49 of Roth (1995).

  12. This reveals a connection between the Nash and strategic theories of bargaining. Given \(\gamma <\hat{{\pi }}\), which in the Nash theory precludes agreement on Bag C, there is no value of \(\theta \) at which agreement on C is the unique strategic equilibrium, whereas there are values of \(\theta \) at which there is equilibrium agreement on each of A and B but not on C. For an extensive discussion of the relationship between these two theories of bargaining see Osborne and Rubinstein (1990, chap. 4).

  13. A full transcript is available at: http://www-users.york.ac.uk/~jdb1/3johns/mk4tscript.pdf.

References

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Correspondence to John D. Hey.

Appendices

Appendix 1

1.1 Instructions

1.1.1 Initial (pre-recorded) oral instructions prior to Part 1

Thank you for participating in this experiment. We hope that you will enjoy it. If you have a mobile phone with you, please check now that it is switched off. [pause]

The experiment requires you to make a few simple decisions which, together with a random factor, will determine the amount you are paid at the end of the session.

There are 16 participants in this session, all facing the same decisions and receiving the same instructions. Beside your terminal you have an envelope, some blank paper, and a pen. Please do not open the envelope until instructed to do so. The pen and paper are provided should you wish to keep a record of your decisions, although it is not necessary to do this. Please leave the pen here at the end of the session.

The session is in two parts. Decisions in Part 1 will determine an amount of money which we will call your dividend. This amount may vary from one individual to another.

However, whether or not you receive your dividend will depend on Part 2, where you will have to agree some decisions with other participants. We will give you further details on this at the start of Part 2.

You will receive instructions both orally, like this, and also on the computer screen. In addition, at all times there will be an information bar at the bottom of the screen. This will remind you what action needs to be taken at that time.

You will have opportunities to ask questions should the instructions not be clear to you. Otherwise, however, you must remain silent throughout the session. At various times you may have to wait for other participants to complete their decisions. If so, please be patient.

Before we proceed to Part 1, are there any questions? [pause]

Please click the Start button now. Read the onscreen information and then wait for further instructions. [pause]

Your task in Part 1 is simply to choose one of the seven boxes. Each box contains £30, to be shared equally among the participants choosing that box. There will be three rounds. The first two are for practice only, and will not count. But the third round is for real, and will determine your dividend.

There will be no further oral instructions until Part 1 is completed. Are there any questions? [pause]

Please make your first practice selection now and then follow onscreen instructions until Part 1 is complete.

1.1.2 (Pre-recorded) oral instructions prior to Part 2

Part 1 is now complete. Your dividend has been computed, but will not be revealed to you until the end of the session.

We will now proceed to Part 2, which consists of four rounds. In each round the computer will pair you, at random, with another participant. It will designate one of you as Yellow and the other as Blue. The pair of you have to agree a decision, which will be explained shortly. You will then be assigned a new partner for the next round, and so on.

Thus, after four rounds, you will have agreed four decisions, each with a different partner. However, only one of these four agreements will actually count for you.

At the end of the session, each participant will be paid individually in private, in the adjoining office. So no other participant will know what payment you receive, unless you yourself choose to reveal it to them afterwards.

Your payment will be determined as follows. Firstly you will draw a number from 1 to 4, from this bag. This will select which of the four rounds in Part 2 is to count for you. Your colour, either Yellow or Blue, will be as designated in your selected round. Then you will draw a ball from this bag, which will contain some yellow and blue balls, and possibly some red balls. If you draw your designated colour, then you will be paid your dividend. Otherwise you will be paid nothing.

We have not yet told you how many balls of each colour will be in your bag. In fact, this is the decision you have to agree with your partner. The contents of your bag will be as agreed by you and your partner in your selected round.

The envelope contains a summary of the information so far. Please open it now and read the summary. [pause]

You may consult the summary again at any time during Part 2.

In each round you will communicate with your partner only via the computer. Instructions for doing this will appear on your screen. Are there any questions? [pause]

Please click the Continue button now. The next few screens give you further details on Part 2, and enable you to practice communicating with your partner. Please note that for the purpose of these practice screens you will be communicating with yourself, as if you were your own partner.

Please read and follow the instructions, continuing through the practice screens in your own time. [pause]

Are there any questions? [pause]

Then please begin Part 2 now.

1.1.3 Written summary information, provided prior to Part 2

Part 1

Your dividend is determined. It will be revealed to you after Part 2.

Part 2

  1. Round 1

    The computer randomly assigns you a partner, and designates one of you as Yellow and the other as Blue. You and your partner agree the contents of the bag.

  2. Round 2

    The computer randomly assigns you a new partner, and designates one of you as Yellow and the other as Blue. You and your partner agree the contents of the bag.

  3. Round 3

    The computer randomly assigns you a new partner, and designates one of you as Yellow and the other as Blue. You and your partner agree the contents of the bag.

  4. Round 4

    The computer randomly assigns you a new partner, and designates one of you as Yellow and the other as Blue. You and your partner agree the contents of the bag.

Payment

You are paid individually and privately in the office, as follows....

You select one round (1–4) at random. Your colour (Yellow or Blue) is as designated in that round, and the contents of your bag are as agreed with your partner in that round.

You draw a ball from your bag. If it is your designated colour, then you are paid your dividend. Otherwise you are paid nothing.

Appendix 2

1.1 Screenshot from experiment

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Bone, J., Hey, J.D. & Suckling, J. What price compromise?. Theory Decis 77, 359–376 (2014). https://doi.org/10.1007/s11238-014-9465-9

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  • DOI: https://doi.org/10.1007/s11238-014-9465-9

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