with the bid the other strategy would have given with the playersÊ actual value. We are trying to find a Nash equilibrium, so s(v i ) must satisfy equation (3). The left hand side expresses the expected utility using strategy s(v i ), and the right hand side the expected utility from using any other strategy,
ିଵ ሺ ݒ
ିଵ ൯ሻ ≥ ݒ ିଵ ൫ ݒ
− ݏ ሺ ݒ ሻ൯ + ሺ− ݏ ሺ ݒ ሻሺ1 − ݒ ିଵ ሻሻ (3)
ݒ
− ݏ ሺ ݒ
ሻሻ + ൫− ݏ ሺ ݒ
ሻሺ1 − ݒ
Cancelling common terms you get:
− ݏ ሺ ݒ
ሻ ≥ ݒ ିଵ ݒ
(4)
− ݏ ሺ ݒ ሻ
ݒ
Then, rewriting the right hand side as ݃ሺ ݒ ሻ = ݒ ିଵ ݒ
ሻ we can find the equilibrium by
− ݏ ሺ ݒ
setting v equal to v i and maximizing g(v) . The resulting differential equation ఋ ఋ௩ ݃ሺ ݒ ሻ = ሺ݊ − 1ሻ ݒ ିଵ − ఋ ఋ௩ ݏ ሺ ݒ ሻ = 0 quickly resolves into
ఋ ఋ௩
ିଵ
(5)
ݏ ሺ ݒ
ሻ = ሺ݊ − 1ሻ ݒ
ሻ = ିଵ
. 3
and thus ݏ ሺ ݒ
(6)
ݒ
We have discovered the Nash Equilibrium.
Equation (6) maps values v i onto bids s(v i ) and expresses the Nash equilibrium we were after. So, for example, if there were two people bidding for an object, the equilibrium strategy that each should employ is to bid ௩ మ ଶ of their value. As values are all in the range [0,1] the equilibrium bid will tail off as the power n increases (as more people join the game). As a very basic model this tells us something quite significant about how hard everybody should end up working for that promotion: for sure, you should never bid an equal amount of effort as your value of the promotion. We must be clear about what exactly we have established. We have not discovered a dominant strategy (optimum strategy no matter what others do). This is obvious: say there were two people bidding for a promotion, which they both valued at a value v . If player 1 bid v, player 2Ês optimum strategy would not be to bid ௩ మ ଶ , that would result in a payoff of − ௩ మ ଶ . Clearly a superior strategy in that case would be to bid 0, with a payoff of 0. Our model is rather simple . Some of the assumptions are unrealistic and there are many other things that have not been considered here. For the rest of this essay I will examine some potential improvements and the intuitive direction these alterations will take our predictions (the formal mathematical proofs are too lengthy and complicated to be included in full). I will also discuss other potential differences between human behaviour and what the assumptions underlying the model, and how to understand these too. LetÊs examine some of the assumptions. Assuming a uniform distribution U[0,1] for the promotion values is unrealistic. True, people will value a job differently: some will be able to translate a job into larger pay than others would, some may enjoy aspects of a job more than
3 Easley, David, and Jon Kleinberg. ÊNetworks, Crowds, and Markets: Reasoning about a Highly Connected WorldÊ. New York: Cambridge UP, 2010. Print.
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