04_Mechanism_Design_with_Money_&_Single-Item_Auctions

• having agents interact through an institution (called center or principal) -> implement social choice function

• types and preferences of differetn agents may be information private to each agent…

• => Principal reuqires truthful information about agents preferences to come to best decisions

• => agents may lie…

• Tuple (𝐼, 𝐴, 𝑂, Θ, 𝐹, 𝑢) where

  • 𝐼 is a set of agents;

  • 𝐴 = 𝐴1 ×···×𝐴𝑛, where 𝐴𝑖 is the set of actions or outcomes;

  • 𝑂 is the set of outcomes;

  • Θ = Θ1 × . . .× Θ𝑛 , where Θ𝑖 is the type space of player 𝑖 ;

  • 𝐹 ∶ Θ ↦ [0,1] is a common prior over types; and

  • 𝑢 = (𝑢1, . . . , 𝑢𝑛), where 𝑢𝑖 ∶ O × Θ ↦ R is the utility function for player 𝑖

• Pair (A,M)

• -> A are actoin sets of players

• M: A -> Δ^𝑂 -> maps each action profile to a distribution over outcomes

• => probability of outcome occuring based on certain probabiliity for each action profile…

• if there is an equilibrium strategy profile of the game induced by M

• such that M results in the outcome of some social choice function

• => basically, if the probabilities of M are in such a way, that each player choses to play so that it reuslts in the social choice funciton C…

• for any vector in types (whatever the type probabilities…)

• the game has an equilibrium in dominant strategies

• and in any such equilibrium a*

• we have M(a*) = C(u(⋅, 𝜃))

=> the result of the mechanism on the dominant strategy equilibrium

=> is the same as the social choice funciton on the known utilities of the types…

=> one on actions (votes) and one on types…

=> outcome of mechanism on action profile is the same as social choice function on types given an bayes nash equilibrium induced by it…

• mechanism so that every bidder reports his type…

• => direct revelation mechanism is incentive compatible if truth telling by all agents is an equilibrium

• => no agent would deviate and not tell the truth…!

• there is never an incentive to lie…

• of for any i , for any type vector, any vector of type reports ß_-i and for any alternative type we have

• => utility of truthful report is at least as high as otherwise…

• For any equilibrium of any arbitrary mechanism

  • => basically for whatever equilibrium in whatever mechanism

• there is an incentive compatible direct mechanism that is equivalent

  • there is a mechanism where the agents have no incentive to lie and report their types that has the same outcome

• Use quasilinear functions

• We redefine outcomes o element O as functoin of x,p

  • where x is a non-monetary outcome (assignments, allocations… (i.e. what hose everyone gets according to i.e. matching))

  • and p is a payment vector

• agent calculate valuation for outcome via valuatoin function (v_i(x,theta (type))

• utility function ui(o,type) = Ui(vi(x,type)-pi)

• -> utility function is function of risk attitude and valuation-price (i.e. valuation - actual price ; charafcterizing attitude and value of money)

• function is quasi linear if it is linear in payment

  • with risk neutral participants: u_i(o,type) = v(x,type)-p_i

  • if U_i monotonic transformaiton -> stil be quasi linear…

• utiility not influenced by amount of money agent has (no money constraint)

• agents dont care how much others are made to pay (no externalities for payment)

• in auction theory, agents can transfer utility via money (transferable utility)

• indepentent private values (IPV) paradigm

• common values

• interdependent values

• v_i(x,type) = v_i(x, type_i)

• => bidder only knows value of object to himself but not of others at time of bidding…

• => knowledg of other bidders value for the object wont affect how much the object is worth to the bidder

• => most plausible when values derived for consumption or use alone..

• value same for all bidders

• => appropriate when value of object derived from market prie that is unknown at time of bidding

• => e.g. tract of land wiht unknown quantity of oil underground

• values depend on own and other bidders values

• e.g. object is asstet that can be resold after auction

• sums utility of each individual to obtain societys overall welfare

• measures social welfare based on

• utility of worst-off member of society

• => min i elemen I (v_i(x, type))

• never an incentive to lie about one’s type (theta, reference Bayes games)

• valuation of object (x(thetas), theta_i) - price of this type vector

  • reminder: x is outcome based on types…

• is equal or greater than any other value that can be reached with other type of player i…

• => strategy proof if

• if for any type, the valution - price is >= 0

• direct quasilinear mechanism

• -> we have reported types (which indicate the valuation…)

• -> we have max price v’_i (outcome (reaiöized price), valuation) -> which is basically the valuation

• outcome is chosen by maximum v’_i -> maximum utility…

• 
  

• o = (x, p) -> x : allocation; p : price

• utility = valuation - payment

• auctioneer calls out low price and raises it in small increments provided there are two or more active biders

• -> auction ends when ther is only one remainng bider

• auctioneer begins calling price suficiently high so no bidder is willing to pay

• price gradually lowered until some bidder indicate it wants to purchase

• first bidder wins auction

• bidder submit bid in sealed envelope

• at pre-determined time, auctioneer opens all enveloopes and ranks bids

• -> highest bidder obtains object and pays price

• bidder submit bids in selaed envelops

• at pre-determiend time, autioneer opens envelops and rank bids

• highest bidder obtains object and pays second highest bid amount

• revenue/expected selling price -> optimal auciton design

• allocative efficiency -> efficient auctoin design

• bid ß(v) = v

• => if smaller bid, and someone bids in between my bid and my valuation, i am in disadvantage, as if i bid higher (still lower than v), i would have higher utiility

• if someone bids lower than me, nothing happens

• if someone bids over me, i loose no utility (as bidding higher would result in negative utility)

• => if higher bid

• -> if s.o. bids higher, nothing changes

• -> if s.o. bids

=> weakyly dominant if bidding v (truth)

• weak domainance

• -> dropout when bid = v

• if regardeless of what other players do -> strategy earns payoff et least as high as any other strategy

• and strictly higher payoff fore some profile of the otehr palyers strategies

• same price and same outcome in both autcion formats

• if informatino is revealed about when bidders rop out

• => may affect individuals estimate of value…

• first price auctoin

• second price auction

• for every strategy in one auctoni format

  • -> there exists equivalent strategy in other auctoin formatn and vice versa

• strategic equivalence => outcome equivalence

• expected value in the lower of n draws

• => E[v_k(n)]

• => k as order, and n is number of draws…

• => THus E[v_2(2)]

• 100 * (n-k+1) / (n+1)

• n grows -> sellers revenue increases

• n to unbound -> seller earns all the surplus..