Games Flashcards
Q
A
What is rationality (pt1)?
A player has beliefs about the other players’ decisions
What is rationality (pt2)?
A player maximises utility based on those assumptions
What is a strict dominating strategy?
A strategy that always outperforms another strategy, no matter my beliefs about other players’ decisions.
Common knowledge
Every player knows that every one knows what everyone knows….
What is a mixed strategy?
A lottery over your pure strategies
Why is Nash Equilibrium problematic (assumption)
We need to know our opponent has a correct belief about our strategy
Indifference curves in the three lottery space
Parallel and straight
How is indifference presented in expected utility notation
Weak preference in both directions
How is strong preference presented in expected utility notation
Weak preference one way, and not the other way
What is completeness (Axiom 1 of expected utility)?
Agents can give you a preference between two lotteries (or be indifferent) for every lottery
What is Transitivity (Axiom 2 of expected utility)?
If A is better than B, and B is better than C, A must be better than C
Theorem of expected utility
If we have a finite number of transitive and complete ranks of lotteries, we can build a utility function
What is Independence (Axiom 3 of expected utility)?
Where lotteries split into multiple lotteries, giving identical probabilities of outcomes, we should be indifferent between this setup in the original
What is Continuity? (Axiom 4 of expected utility)
If we have three lotteries, we can make an indifference condition through a lottery between the higher and lower preference utilities
What is the use of an extensive form game
When decisions happen over different periods of time, and people have dynamic information
What is the history of a game?
The sequence of decisions made to get to a certain point
What are the two types of histories
Terminal and Subhistories
How do you present a normal form (non table)
All strategies possible for each player listed
Compromise between NE and backwards induction
Subgame perfect NE
What is Zermelo’s theorem
In a two-player, finite game, with strict preferences win > draw > lose, there is a subgame perfect Nash equilibria
Rubenstein Bargaining model
Infinite bargaining setup where payoffs are discounted at a consistent rate each period
Critique of Independence of Irrelevant Alternatives
In bargaining situations, removing threats can lead to changes in solution
What is the Jury setup
Actions (acquit/convict), Types (looks innocent, looks guilty), State (Innocent, guilty), Payoffs (correct result gets 1, incorrect is 0)
What is Bayes rule?
The probability of A given B is the probability of A and B, divided by the probability of B
What is a bayesian nash equilibrium?
Everyone is maximising their own utility given what the other player’s actions are and their beliefs about other player’s types
In what scenario does my vote matter in the voting problem?
When the vote numbers are equal
What is the word for a voter who casts the deciding vote?
Pivotal
What is the strategy in the voting game
Voting when their cost is low enough and the probability of being pivotal is high enough
As n increases, what happens to the probability of being pivotal?
Decreases
What do people vote if they get the guilty signal?
Vote to convict
What do people vote if they get the innocent signal?
Some randomisation over innocent and guilty
What is a Weak Perfect Bayesian Equillibrium?
Players estimate probabilities of the “nature” at each information set, which we have defined using bayes rule, and then maximise utility with these beliefs
What are the types of equillibria in the signalling game?
Pooling and seperating
In the pooling equilibrium, what are the beliefs about information sets?
Just nature
Stage 1 - Pooling Equilibrium Checklist
Make a hypothesis on the sender equilibrium strategies
Stage 2 - Pooling Equilibrium Checklist
Given the hypothesised strategies, work out the receiver’s beliefs on the equilibrium path
Stage 3 - Pooling Equilibrium Checklist
Compute the receivers best reply on the equilibrium path
Stage 4 - Pooling Equilibrium Checklist
Make sure that our original equilibrium strategies are the best reply for each type of sender
Stage 5 - Pooling Equilibrium Checklist
Find the restrictions that step 4 imposes and the receiver’s off-equilibrium beliefs.
Stage 1 - Separating Equilibrium Checklist
Make a hypothesis on the sender equilibrium strategy
Stage 2 - Separating Equilibrium Checklist
Identify the receivers beliefs at every point on the equilibrium paths (all are in this case)
Stage 3 - Separating Equilibrium Checklist
Compute the receiver’s best reply in each of these cases
Stage 4 - Separating Equilibrium Checklist
Verify that the sender is acting rationally to nature
What is herding?
When your signal is dominated by external signals, so everyone goes with the same outcome in social learning
What is the payoff in the cheap talk game?
Negative deviation squared
How does Bayesian Persuasion differ from cheap talk
The sender commits to a communication strategy
What structure does the solution to Bayesian persuasion come in
The sender maximises the chance of sending the “2” signal while keeping his signals believable enough to be trustworthy
Stable pairing
When there is no alternative that both parties would prefer
Deferred acceptance procedure
All A propose to their favourite B, all B reject everyone except favourite A, continued until all gone
What is an auction - theory definition
Allocation mechanism of a scarce good with transfers
Structure of an auction - setup
The auctioneer communicated how the “messages” will be translated into an allocation and a transfer
Structure of an auction - messages
All players send a message into the message space
When are auctions a bad idea
Low commitment, low number of bidders, high bidding costs
Structure of an auction - solution concept
All individuals’ bidding strategies
What are the impacts of transfers on total utility?
Nothing - transfers are irrelevant
What is Ex-ante
No one knows their type
What is interim
You know your own type but not everyone’s
What is ex-post
You know everyone’s type
What is a direct mechanism?
A bidding mechanism that bidders truthfully report their types
What is the revelation principle
We can assume every mechanism can be presented as a direct mechanism
Reasoning behind the revelation principle
You are always able to avoid the intermediate step of message mapping by going directly from types to transfer and allocations
What is the goal of VCG?
Internalising everyone’s utility into the transfers of the individual
Benefit of VCG
Bidders maximise their own utility by bidding truthfully
What is the pivot mechanism?
The payment is set at the externality inflicted on the other bidders
What is an example of a pivot mechanism?
Second price auction
What is shading?
The difference between your personal value and the bid
Assumption about type distribution
Everyone has identical type distribution ex-ante, which gives us symmetric strategies in the interim (as no bayesian update based on type)
How does the level of shading change as number of bidders changes?
Shading falls
On the uniform distribution, how much shading do we do?
1/N
Which gets more revenue - first or second price auction?
Same
What is the formula for integration by parts?
UV - INT(UV’) = INT(U’V)
Structure of the advertising market
Advertisers bid on ad exchanges, second highest bid is submitted to the ad network, who choose the highest of those bids
Is the advertising market ex-post efficient?
No, as someone could have a higher value but lose because of their second-highest bid, which they can’t control
What did google do that messed up the advertising market?
Had both an ad exchange and ad network, and then allowed the exchange a “last look advantage.
Result of the google case
Anti-trust case ended the second price auctions
Stages of solving the differential equation
Homogenous problem, varying the constant, verifying the constant (requires the initial)
Indirect utility approach
Using our type, what is the maximum utility we can get - our willingness to pay to get into the auction
Rule from indirect utility
Value of higher type is equal to the value of the increased interim probability of winning the auction
How to set up indirect utility solution
Utility is both equal to the classic description of winning utility, as well as the integral of the marginal probability of winning over price, integrated in the range - set them in a fraction
How do we find the Kth highest bid?
Order statistics
Order statstic expression
Probability of k-1 higher, n-(k-1) lower, and binomial multiplier
Model - multiple mechanism designer
Both designers offer some subset of A, B, C, and then the chooser maximises their equillibrium
Result from the multiple mechanism design
Using out-of-equillibrium messages can allow us to manipulate the equillibrium
Requirements for incentive-compatible direct mechanisms
Probability of winning is non-decreasing in type, and utility the utility benefit of the higher type is only based on increased probability of winning
Proof of incentive compatibility
We must benefit by bidding truthfully - assuming this, setup an inequality and rearrange, find that the probability of winning has to be non decreasing, and the marginal gains of in W are equal to the increased prob of winning.
Revenue equivalence theorem
If two auctions have the same expected payoff from their lowest valuation, and the same probability of winning ex ante (symetric), then the auctions will have the same expected revenue
What is another way of saying expected auction revenue
The sum of all the expected transfers of the players - revenue equivalence is maximised by setting U=V
What is the difference between utility (U), and forced utility (V),
V is unoptimised bidding - in a direct mechanism, they are equivalent
Structure of the wind energy subsidy
Government loans seabed space, signs contracts to pay the extra “strike price” for a certain amount of energy produced (years), the companies build wind power plants
What does the auctioneer choose
The “q” value - the interim probability of winning
Definition of the symmetric revenue maximising auction
Assigns the object to the highest bidder, conditional on the “virtual surplus” being positive
Finding on reserve price
Any standard auction with a common reserve price is revenue maximising
Hetrogenous ex ante
Bidders have different probabilities of winning
Results of “weak and strong bidding”
Weak bidders will bid higher with the same valuation
Issue with weak bidders bidding more aggressively?
Ex-post inefficiency in first-price: weak bidder can win the item with a lower valuation
What gives higher revenue in an asymmetric bidding situation
First price gives higher than second price, due to its ex-post inefficiency
Second-price auction - asymmetric auction bidding strategies
Same as always, bid your type (same outcome as symmetric)
Why can auctioneer increase revenue in the asymmetric auction
Surplus from weak bidder is higher, so the auctioneer can favour them, forcing the strong bidder to bid harder. This is the first price auction
True optimum auction for revenue optimisation - asymmetric
Asymmetric reserve prices, due to different virtual surplusses
Expression for virtual surplus
Theta - (1-F(theta)/f(theta))
What is a bidding ring?
Collusive agreement between bidders before the actual auction
Issue with bidding rings
Enforcing commitment to not bid in the final auction
Issue with transfers in a bidding ring
People may participate with no intention of winning to get the money, also easier to trace
The result of no commitment or transfers in the bidding ring
Cheap talk, result should be the same as no bidding ring
Result about bidding rings with commitment
The best utility result is randomising the winner - no one reveals anything about their type, and you are best of randomising
Informed-uninformed bidder equilibrium - 2nd price
Informed bids type, the uninformed either bids maximum and pays value, or bids nothing and the other person wins
What is the winner’s curse?
When you win an auction, information about other’s assessment of value is revealed to be lower
How do we know that the uninformed bidder randomises?
To avoid winner’s curse, if they bid a specific number, they only win when they’re taking a loss in the NE
Proof that both bidders have the same range of bids in equillbrium
Must have same upper bound, as otherwise the other bidder can reduce max at no cost, must have the same lower bound, as if theta is 0, one with the lower minimum gets a negative payoff, or if their minimum is higher than the other, can benefit by reducing
Proof of uninformed bidder having 0 indirect utility
Indifferent between all bids (as randomised), and one of those bids is 0 at 0, so all must have 0 indirect utility
How does the non-neighbour choose their bid in the oil drilling auction
“Makes up” a fictitious oil amount, uses other bidders bidding strategy
Distribution of bids in the non-symmetric info case
Identical from neighbour and non-neighbour
Common values
All see a signal, but the valuation is derived from everyone’s signal
How do you conceive your bid in a common values setting
Value of item subject to your bid, and conditional on everyone else having a lower signal
What yields a higher revenue - first or second price auction, in a commons value situation
Second price - bidders are less worried about being overly optimistic, as would be insured by the second highest bidders. This leads to more optimistic bidding
