Static games incomplete info Flashcards
The sealed-bid first price auciton, how is it incomplete, and why do we need to use BNE instead of NE?
Players simultaneously bid for a good, not as in a normal auction in real life. The highest bidder wins, pay the highest price. Risk neutral
The incomplete part of this game, is that bidder I don’t know bidder j´s valuation. Therefore we have to use BNE, because our expectation is based upon beliefs about the other players valuation. expected payoff.
What is a Baysian Nash Equlibrium, when we look at it next to the casual NE.
We use a BNE when we have incomplete information, which introduce a new layer to the casual NE.
The NE under complete information
- Players
- Actions
- Payoffs
The BNE under incomplete information
- Players
*Types
*Common prior beliefs.
- Actions
- Payoffs
In BNE we are now facing different kinds of types of players and we have uncertainty of which type we are facing. But we have the same idea that, if we find the a place where both players have BR, then it is a BNE.
What is the cumulative distribution function (CDF) for uniformly distributed variabel
It is P(X<x)=(x-a)/(b-a) for X\sim(a,b)
so if we had a uniformly distrubution like, v_i\simu(1,3), then a=1 and b=3 and we would find x in the function.
The sealed-bid second-price auction (Vickrey auction). Why is it smart
Players simultaneously bid for a good
Highest bidder wins, but pay the second highest price. So pay the amount the person just before bid. Risk neutral
Smart because:
1. Uses weaky dominating strategies.
2. Strategy proof. Normally we think about, what you should do considering the other player, and him considering you…. Here we don´t need that. You write down your true value, no more than that. No strategizing.
3. Honest. No one wants to deviate, because of the structure of the game. You don´t need to lie, bluff and so on.
Works with incomplete information, again because everybody just write down there true value.
What is the winners curse? How does it relate to that the seller gets higher expected payoff, when the there is more players.
That you win, but you paid way more than you wanted for the price. Your bid is higher than your valuation, v_i-b_i.
The theory tells us that with more players the price will increase, which means that the expected payoff for the seller increase ass well.
What does it mean when we have to find a symmetric BNE?
It means we only have to find a BNE for 1 of the bidders, since they are drawn from the same distribution.
Describe this in words u_i=v_i-b_i, and tell why it is timed on equation in a generel problem.
That is the utility given by the bidder i´s valuation minus his bid. We time the acutual payoff of bidder i on the probability that he wins over b_j.
v_i can also be interpreted as a bidders maximum willingness to pay.
If the lowest-type bidder will place a bid equal to his valuation, whereas all other bidders will bid strictly less than their valuation, and comment briefly on why this is the case. Does this relate to the idea of the ‘winner’s curse’?
Solution: notice that b(vi) = vi + 3 implies b(vi = 3) = 3 and b(vi) < vi for all v ∈ (3,4]. A bidder with valuation v ∈ (3,4] will engage in bid-shading, by bidding less than her valuation. Bidding her valuation would give a payoff of zero, even if she wins the auction, whereas bidding lower can give her the possibility of winning the auction and earning a strictly positive payoff. In contrast, a bidder with valuation v = 3 is willing to bid her valuation; a higher bid clearly does not make sense, as it can possibly lead to losses, whereas a lower bid is not attractive because it would never allow her to win the auction. This idea of bid shading is unrelated to the winner’s curse. The key point is that the winner’s curse applies in a common value setting, where bidders should take into account that winning the auction constitutes ‘bad news’ about the value of the item in question; whereas here we look at a private value setting, where such concerns are not present.
