L10 - Game Theory: Mixed Strategies II

Question 1

How would you solve the below game?

Answer 1

1. if you are not sure, move onto Step 2b anyways, you’ll get a nonsense answer if there is not a mixed eqm

Question 2

What are all-pay auctions?

Answer 2

• All pay auction –> In economics and game theory, an all-pay auction is an auction in which every bidder must pay regardless of whether they win the prize, which is awarded to the highest bidder as in a conventional auction.
• There are many economic situations where players compete with sunk resource investments in order to win some form of prize.
• These costly investments must be paid regardless of whether they win or lose.
  
  • • Political Campaigns
  • • Lobbying
  • • Contract tendering
  • • R+D
  • • Rewards in Organizations (e.g. promotions)
  • Legal Battles
  • Sports
  • Military

Question 3

How do you set up an all-pay auction game?

Answer 3

Question 4

How do you solve the model of an all-pay auction game?

STEP 1

Answer 4

• Wouldnt pay 100 to win a prize of 10 thus wouldn’t bid above the prize V

Question 5

How do you solve the model of an all-pay auction game?

STEP 2a

Answer 5

• s there a mixed eqm? Probably yes, as there is no pure strategy eqm. At any possible outcome in pure strategies, at least one player would want to deviate. Why?
• This goes back to the logic that we saw in the classroom….
  
  • E.g. Suppose both players set b=V. Each player would earn (V/2)-V<0 and would strictly prefer to set b=0 to guarantee 0 instead.
  • E.g. Suppose both players set b=0. Each player would earn (V/2)-0>0. However, any player would strictly prefer to deviate by setting b=0.01 to win outright and earn V-0.01 instead, because V-0.01>(V/2) (if V is suff large).
  • Similar arguments can be made for any possible outcome in pure strategies

. Related arguments can also be made to show there can never be any ties within an equilibrium. If there were, someone could always optimally deviate

Question 6

How do you solve the model of an all-pay auction game?

STEP 2b

Answer 6

• Now, we need to start calculating the mixed strategy equilibrium. To do that, we need to consider a probability distribution over actions.
• Given the continuous action space between 0 and V, we can’t assume the players simply play over different actions with probability q and (1-q) again.
• Instead, we must allow that the players to potentially play overall actions (bids) in between 0 and V with a cumulative distribution function, F(b) –> where F(b) = Pr(bid ≤ b) . This ‘cdf’ describes the probability with which a player will play different bids in our continuous strategy setting.

To be well-behaved here, a cumulative distribution function should have:

• • F(b)=0 when b=0 (Prob of bidding less than or equal to zero is zero)
• • F(b)=1 when b=V (Prob of bidding less than or equal to V is one)
• • F(b) is never decreasing
  
  • (E.g. The prob of bidding less than or equal to 3 can’t be less than the prob of bidding less than or equal to 2.)

*** if one bid has the expected payoff 0, and I’m indifferent between bids from 0 to 10, then all should have the expected payoff 0 regardless of what is bid

• assume all assumptions used so far!

Question 7

What is the economic importance of the equilibrium derived from the all-pay auction game?

Answer 7

• The equilibrium provides a general prediction across the family of all-pay auction situations: political campaigns, contract tendering, R+D…
  
  • • The value of the winning bid is random.
  • • Even with two players, competition is strong enough that both players receive an expected payoff of zero. The costs of bidding offset any possible prize.
  • • Related models can also explain mixing over prices and advertising decisions by firms, in ways that are consistent with sales behaviour.
  • • The basic model can be extended in numerous ways: e.g. more players, and asymmetric players

In experimental tests of these situations, subjects typically mix in ways that are close to that predicted, but they often:

• o Take an unusually long time to learn/converge towards the NE,
• o Place too much weight on very low bids or very high bids, and
• o Display aggressive over-bidding as consistent with gaining some additional utility from ‘winning’

Question 8

What is the equilibrium effect of more players in the all-pay auction game?

Answer 8

• As the number of players increases, F(b) increases at any 𝟎 < 𝒃 < 𝑽.
• This means that players place more probability on lower bids.
• Implies that as n increases, average bids actually fall!
• Contrasts to the standard idea of more players increasing competition
• . Instead, as more players join, players reduce their bids because they realise their chance of winning falls. In some sense, they gradually give up a bit!
  
  • opposite or Cournot, Bertrand where if more people join a market they compete harder
• This is consistent with several intuitive patterns of behaviour, such as increased youth apathy and school drop-out under globalisation