Job promotion competitions
Josh Robinson If asked the question, many people would say that one should work as hard as possible in order to secure a promotion. An economist when confronted with this question may correct the previous assertion by suggesting that one should only work up to the point where the cost of the extra work required to gain the promotion equals the extra benefits gained from it. However, I would suggest a slightly different approach. Auction theory, a branch of game theory, allows us to assess what rational ÂplayersÊ should do given other competitors actions. The good (?) news is that auction theory tell us that you shouldnÊt be working as hard as possible for that job. Parents (including mine) may be somewhat reluctant to embrace this suggestion, so I shall try and explain why. Job promotion ÂcompetitionsÊ are tough: you have to work and put in effort in order to try and achieve that promotion. There is, however, a good chance that the promotion will be given to someone else, maybe because they worked harder, or because they are simply better at their job (perhaps this Ânatural skillÊ is a dividend of past efforts). So, you should put in as much effort as possible in order to try and win, right? Not necessarily. The potential to lose means that putting in maximum effort could maximize your chances of winning the promotion, but your expected utility -all potential outcomes weighted by their probabilities - is unlikely to be maximized. A promotion competition can be modelled as an Âall-payÊ auction: an auction where all bidders pay their bid, regardless of whether they win or not. We want to map ÂvaluesÊ, v i , to ÂbidsÊ, s( v i ) . LetÊs assume that there are n players whose values of the job are independently and uniformly distributed between 0 and 1, and that s(v i ) is an increasing function, i.e. as your value v increases, so too does your bid. Using standard probability theory, the Expected payoff for player i is
(1)
ሺ ݒ
ሻ =
ሺ ݒ
ሻሻ
ܷ
− ݏ ሺ ݒ
ሻሻ + ሺ− ݏ ሺ ݒ
ሻሺ1 −
Where p i is the probability of player i winning. The first part models the payoff from winning; the second part shows the payoff from losing. But because we have modelled the values as a uniform distribution [0,1] we can find p i precisely. The probability of any bid being lower than v i is itself v i , therefore = ܲ൫ ݒ < ݒ ൯ = ݒ ିଵ , where ݒ is the highest of all bids other than v i . So,
ିଵ ሺ ݒ
ିଵ ൯ሻ
(2)
ሺ ݒ
ሻ = ݒ
ܷ
− ݏ ሺ ݒ
ሻሻ + ൫− ݏ ሺ ݒ
ሻሺ1 − ݒ
Equation (2) is the same as (1), but with ݒ . We would like to find a dominant strategy: a strategy that is superior to any other no matter what bid is made by anyone else. But, in reality the best we can do is to search for a Nash Equilibrium 1 : ÂA pure-strategy Nash equilibrium is an action profile with the property that no single player can obtain a higher payoff by deviating unilaterally from this profile.Ê 2 In this case we will find the equilibrium strategy for all players. This is known as a symmetrical equilibrium because every player is playing by the same rules, and therefore uses the same strategy (though probably not the same bid values). But surely other strategies than the specific strategy s(v) could be used? We can, instead of specifying many different strategies, input ÂfakeÊ values v into the same strategy, which comes out ିଵ substituted in for p i
ሻ〉 is a profile of actions ܽ ∗ ∈ ܣ
1 Mathematically expressed: ÂA pure Nash equilibrium of a strategic game, 〈ܰ, ሺ ܣ ሻ, ሺ ݑ Ê . http://www.econ.brown.edu/fac/Kfir_Eliaz/206_notes_nash.pdf, (date accessed 30.07.13). 2 http://www.columbia.edu/~rs328/NashEquilibrium.pdf (date accessed 29.07.12). with the property that for every ݅ ∈ ܰ , ݑ ൫ܽ ,∗ , ܽ ି ∗ ൯ ≥ ݑ ሺܽ , ܽ ି ∗ ሻ for all ܽ ∈ ܣ
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