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Take, for instance, that you have a game similar to the one proposed by Levitt and Duggan (2000) in sumo wrestling (link: http://www.nber.org/papers/w7798.pdf, section II (pages 9-10) in which the paper describes the model).

Summarizing, a fighter maximizes a expected return in a non-rigged match. For a given fighter consider taking the bribe, the other minimizes a side payment (bribe) to the point of indifference between taking and not taking the bribe -- so, why would the fighter setting the side payment would not set the bribe so it would be marginally better to accept the bribe than not taking it? Why do the authors include the point of indifference among the possible solutions?

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  • $\begingroup$ I changed the title because I believe this is more general and it reflects your actual question better. If you don't like it feel free to change it back. $\endgroup$ – Giskard Dec 19 '15 at 13:44
  • $\begingroup$ I agree with the change in the title. $\endgroup$ – John Doe Dec 20 '15 at 0:06
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Because otherwise you would not have subgame perfect equilibria. Suppose I have an item you need and the item is worthless to me. I would be happy to give you the item for any positive amount of money. If you give me no money for it, I am indifferent, both giving you the item and not giving you the item is an optimal choice for me. Just to be clear: we are not bargaining. You set I price and I will take it or leave it.

If you offer price 0 and I give you the item that is an equilibrium. No one could have made a better choice given the other person's choice.
If you offer price 0 and I do not give you the item that is not an equilibrium, because I would give you the item for any small positive payment and you would be willing to pay that, so you could have made a better move.
If you offer price $2a>0$ I will give you the item. But this is not an equilibrium. You could also have offered me $a$, which is also positive, and I would also have given you the item then. So you could have saved some money, your original move was not optimal.

Because there is no smallest positive number, the only equilibrium in such cases is when one side is indifferent.

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  • $\begingroup$ This supposes no negotiation power whatsoever from the seller side, which should be willing to sell at a gain if he's allowing the buyer to make a profit. $\endgroup$ – VicAche Dec 19 '15 at 13:48
  • $\begingroup$ @VicAche Indeed it does. It is an assumption frequently made in competitive markets. I also point out said assumption when I say that there is no bargaining. $\endgroup$ – Giskard Dec 19 '15 at 14:11
  • $\begingroup$ @VicAche From the bottom of page 9 of the article referred by the OP: "Because we assume that the wrestler offering the bribe has all of the bargaining power"... $\endgroup$ – Giskard Dec 19 '15 at 14:19
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    $\begingroup$ you may also want to add that you focus on subgame perfect equilibria. otherwise, offering 2a and accepting all offers equal to or above 2a is a nash equilibrium. $\endgroup$ – HRSE Dec 20 '15 at 5:26
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    $\begingroup$ @denesp, I forgot to thank you for your references o/ $\endgroup$ – John Doe Jan 11 '16 at 20:17
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One thing that goes towards indifference happening in economics is granularity. In simple terms, making a marginally better offer to someone is not enough to offset preferences away from indifference as indifference is a zone of payment (say, $[a-\epsilon, a+\epsilon]$, with $\epsilon$ the granularity of the agent) rather than a single payment.

See for example bottom of page 8 here in Consistency, Heterogeneity, and Granularity of Individual Behavior under Uncertainty, Choi et al., 2007

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  • $\begingroup$ While granularity is certainly interesting, how does this answer the question? $\endgroup$ – Giskard Dec 19 '15 at 14:14
  • $\begingroup$ @denesp because it renders marginal increments useless, thus answering the question: setting a "marginally" better payment is not feasible $\endgroup$ – VicAche Dec 19 '15 at 14:23
  • $\begingroup$ So why not set a payment that is discretely better? Say $a + \epsilon$? I think the OP wants to know why you can be sure of acceptance with indifference. $\endgroup$ – Giskard Dec 19 '15 at 14:26
  • $\begingroup$ @denesp if he makes a discretely better payment he has to choose to do so, at a distinguishable cost for himself (supposing comparable granularities for both agents), so it represents a different situation from the one presented in the provided paper. $\endgroup$ – VicAche Dec 19 '15 at 14:28

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