L10 - Game Theory: Mixed Strategies II Flashcards
How would you solve the below game?
![](https://s3.amazonaws.com/brainscape-prod/system/cm/362/821/425/q_image_thumb.png?1636487748)
- if you are not sure, move onto Step 2b anyways, you’ll get a nonsense answer if there is not a mixed eqm
![](https://s3.amazonaws.com/brainscape-prod/system/cm/362/821/425/a_image_thumb.png?1636487805)
What are all-pay auctions?
- 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
How do you set up an all-pay auction game?
![](https://s3.amazonaws.com/brainscape-prod/system/cm/362/822/412/a_image_thumb.png?1636488179)
How do you solve the model of an all-pay auction game?
STEP 1
- Wouldnt pay 100 to win a prize of 10 thus wouldn’t bid above the prize V
![](https://s3.amazonaws.com/brainscape-prod/system/cm/362/822/667/a_image_thumb.png?1636488442)
How do you solve the model of an all-pay auction game?
STEP 2a
- 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
How do you solve the model of an all-pay auction game?
STEP 2b
- 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!
![](https://s3.amazonaws.com/brainscape-prod/system/cm/362/823/731/a_image_thumb.png?1636489030)
What is the economic importance of the equilibrium derived from the all-pay auction game?
- 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’
What is the equilibrium effect of more players in the all-pay auction game?
- 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
![](https://s3.amazonaws.com/brainscape-prod/system/cm/362/825/426/a_image_thumb.png?1636489860)