Games With Incomplete Information: Theory of Auctions And ...hrtdmrt2/Teaching/GT_2017_19/L11.pdfGames With Incomplete Information: Theory of Auctions And Other Examples Carlos Hurtado

Games With Incomplete Information: Theory ofAuctions And Other Examples

Carlos Hurtado

Department of EconomicsUniversity of Illinois at Urbana-Champaign

[email protected]

July 4th, 2017

C. Hurtado (UIUC - Economics) Game Theory

mailto:[email protected]

On the Agenda

1 Formalizing the Game

2 Soving Bayesian Games

3 Sealed Bid (First-Price) Auction

4 Vickrey (Second-Price) Auction

5 Exercises


Formalizing the Game

On the Agenda





5 Exercises




I Formally, we can define Bayesian games, or "incomplete information games" asfollows:

- A set of players: I = 1, 2, · · · , n

- A set of States: Ω e.g. good or bad car.

- A signaling function that goes into type space and is one-to-one: τi : Ω→ Ti .

- Pure strategies that are profile of actions conditional on player’s type: σi : Ti → Ai

- Individual Utility of outcome y , given actions (a1, a2, · · · , an):

Ui (σ1, · · · , σn|ti ) =∑

a1

· · ·∑

an

ui (y , σ1(t1(ω)), · · · , σn(t1(ω))) · pi (y |ti (ω))

I What would be the BR of player i? maxσi Ui (σ1, · · · , σn|ti ) (not solvable)

I Player i needs to know what −i knows about him. Also, Player i needs to knowwhat i knows about −i . Moreover, i needs to know what −i know about himconditional on what i know about −i , and so on

C. Hurtado (UIUC - Economics) Game Theory 1 / 16


How do we model Bayesian games?

I In order to analyze these types of games, we rely on a fundamental (andNobel-prize winning) observation by Harsanyi (1968):

I Games of incomplete information can be thought of as games of complete butimperfect information where nature makes the first move (selecting T1, · · · ,TN ),but not everyone observes nature’s move (i.e. player i learns Ti but not T−i).

I Harsanyi (1968) doctrine: There is a prior about the states fo the nature that iscommon knowledge

I This is also known as the common prior assumption

I This is a very strong assumption, but very convenient because any privateinformation is included in the description of the types.




I A Bayesian pure strategy for player i in a Bayesian game is a function σi (Ti ), ordecision rule, that gives the player’s strategy choice for each realization of his typeTi . That is, σi : Ti → Ai .

I Player i ’s expected payoff given a profile of Bayesian pure strategies for players(σ1(T1), · · · , σN(TN)) is given by:

ui (σi (Ti ), σ−i (T−i )) = ET [ui (σi (Ti ), σ−i (T−i ),Ti ,T−i )]

DefinitionA Pure Strategy Bayesian Nash Equilibrium (PSBNE) is a profile of decision rules(σ1(T1), · · · , σN(TN)) such that

ui (σi (Ti ), σ−i (T−i )) ≥ ui (σi (Ti ), σ−i (Ti ))

for all σi (Ti ).



Sequential Equilibrium

I Bayesian Nash equilibrium is a straightforward extension of NE:

I Each type of player chooses a strategy that maximizes expected utility given theactions of all types of other players and that player’s beliefs about others’ types.

I A pair (σ, µ) of strategy profile σ and a belief assessment µ is said to besequentially rational, if and only if, at each information set the player who is tomove maximizes his expected utility (BNE)

I Given (σ, µ), a belief assessment µ is said to be consistent with µ iff theconditional probabilities derived by Bayes rule are probabilities given by σ on allinformation sets.

I A pair (σ, µ) of strategy profile σ and a belief assessment µ is said to be asequential equilibrium if (σ, µ) and is sequentially rational and µ is consistent withσ.

I Remark: SE is closely related to another solution concept, called WPBNequilibrium. Sequential equilibrium is a better defined solution concept, and easierto understand. The two solution concepts are equivalent in the games consideredhere.


Soving Bayesian Games

On the Agenda





5 Exercises



Solving Bayesian Games

I Two firms jointly share their research outputs. They might be envisioned asdivisions of the same firm. Each firm can independently choose to spend c ∈ (0, 1)to develop the "zigger", a device that is then made available to the other firm.

I Firm i ’s type is θi , which is believed by firm −i to be independently drawn fromthe uniform distribution on [0, 1].

I The benefit of the "zigger" when the type is θi is θ2i .

I The timing is: the two firms privately observe their own type. Then they eachsimultaneously choose either to develop the "zigger" or not.

I Solve for the Pure Strategy Nash Equilibrium.




I value of the zigger to firm i : θ2i

I payoff if the zigger is not provided: 0I payoff if it builds the zigger: θ2

i − cI payoff if it does not build the zigger but firm −i does: θ2

i

I si : [0, 1]→ yes, noI Let p−i denote the probability that firm −i produces the zigger, given its strategy

s−i .I Firm i should provide the zigger only if

θ2i − c ≥ p−i · θ2

i

I Equivalently,

θi ≥√

c1− p−i

I Hence, firm i and −i use a cutoff strategy.



Solving Bayesian GamesI Let θ∗i be the cutoff point. Then, firm i will provide the zigger with probability

pi = 1− θ∗i = 1−√

c1− p−i

= 1−√

cθ∗−i

I Thereforeθ∗i =

√cθ∗−i

I That is,θ∗2i · θ∗−i = c

I and symmetrically,θ∗2−i · θ∗i = c

I Canceling,θ∗i = θ∗−i

I That is, the only PSBNE is symmetric.C. Hurtado (UIUC - Economics) Game Theory 7 / 16


Solving Bayesian GamesI Substituting into the equation above:

θ∗i = θ∗−i = c13

I Where is the cost of free riding?I The zigger should be provided if

θ21 + θ2

2 ≥ c

I Given that c ∈ (0, 1), we have that c 12 < c 1

3




I Substituting into the equation above:

θ∗i = θ∗−i = c13

I Where is the cost of free riding?I The zigger should be provided if

θ21 + θ2

2 ≥ c

I Given that c ∈ (0, 1), we have that c 12 < c 1

3


Sealed Bid (First-Price) Auction

On the Agenda





5 Exercises




I In a sealed bid, or first price, auction, bidders submit sealed bids b1, ..., bN . Thebidders who submits the highest bid is awarded the object, and pays his bid.

I Under these rules, it should be clear that bidders will not want to bid their truevalues. By doing so, they would ensure a zero profit.

I By bidding somewhat below their values, they can potentially make a profit someof the time.

I Model:- Bidders i = 1, ...,N- Bidder i observes a "signal" θi ∼ F (·)- Bidders’ signals are in [θ, θ] for all bidders.- Bidders’ signals are independent- Bidder i ’s value of the object is vi (θi ) = θi- Bidders’ bids depend on the "signal" θi with the same functional form b(·)

I If Bidder i wins the auction he has an utility of vi (θi )− b(θi )I The probability of wining the auction is

Pr [b−i = b(θ−i ) ≤ bi = b(θi )] = Pr [θ−i ≤ b−1(bi )]




I To ensure that b−1(·) exist, we impose that each bidder uses a bid strategy that isa strictly increasing and continuous.

I Then, the bidder i ’s expected payoff, as a function of his bid bi and signal θi is:

U(bi , θi ) = (vi (θi )− b(θi )) · Pr [θ−i ≤ b−1(bi )]

I that can be reduced to (why?)

U(bi , θi ) = (θi − bi ) · F n−1[b−1(bi )]

I Bidders would like to optimize their bid to maximize their expected utility:

∂U(bi , θi )∂bi

= (θi −bi )·(n−1)·F n−2[b−1(bi )]·f (b−1(bi ))·1

b′(b−1(bi ))−F n−1[b−1(bi )] = 0

I Here we need to assume further that b(·) is differentiable.




I This can be reduced to the following differential equation (why?):

b′(θi ) = (θi − b(θi )) · (n − 1) ·f (θi )F [θi ]

I The solution to this differential equation is the symmetric equilibrium.I Another aproach:

U(b(θi ), θi ) = U(θi ) = (θi − b(θi )) · Pr [θ−i ≤ θi ]

I Because i is playing a best-response in equilibrium:

U(θi ) = maxbi

(θi − bi ) · F n−1(b−1(bi ))

I Applying the envelope theorem

dU(θi )dθi

= F n−1(b−1(b(θi ))) = F n−1(θi )




I Then,dU(θi )dθi

= F n−1(θi )

I imply

U(θi ) =

θi∫θ

F n−1(θ)dθ + K

I Given that U(θ) = 0 we have K = 0.I Then,

U(θi ) = (θi − b(θi )) · F n−1(θi ) =

θi∫θ

F n−1(θ)dθ

I Hence:

b(θi ) = θi −

∫ θiθ

F n−1(θ)dθF n−1(θi )


Vickrey (Second-Price) Auction

On the Agenda





5 Exercises




I In a Vickrey, or second price, auction, bidders are asked to submit sealed bidsb1, ..., bN .

I The bidder who submits the highest bid is awarded the object, and pays theamount of the second highest bid.

TheoremIn a second price auction, it is a weakly dominant strategy to bid one’s value, bi (θi ) = θi

Proof:I Suppose i ’s value is θi , and she considers bidding bi > θi .I Let b denote the highest bid of the other bidders j 6= i (from i’s perspective this is

a random variable).I There are three possible outcomes from i ’s perspective:

1) b > bi > θi2) bi > b > θi3) bi > θi > b




Proof:

I case 1 or 3: Player i would have done equally well to bid θi rather than bi > θi .(why?)

- In case 1: b > bi > θi , she won’t win regardless.

- In case 3: bi > θi > b, she win independently by bidding bi or θi

I case 2: bi > b > θi

- she will win, and will pay b regardless.

- Player i will win and pay more than her value if she bids b, something thatwon’t happen if she bids θi

I Thus, i does better to bid θi than bi > θi . A similar argument shows that i alsodoes better to bid θi than to bid bi < θi


Exercises

On the Agenda





5 Exercises


Exercises

Exercises

I Consider an economy in which there are two consumers, a public good x and aprivate good y . Consider the voluntary contribution game in which each player icontributes an amount zi of his private good whereupon an amount x = z1 + z2 ofthe public good is produced. Consumer i has large initial endowment yi of y and autility function

ui (x , yi ) = aix −x2

2 + yi

where yi = yi − zi . Consumer i ’s preference parameter ai is either 4 or 6, but onlyconsumer i knows which. The other consumer believes that ai = 4 with probability1/2 and ai = 6 with probability 1/2. Compute a Bayesian Nash equilibrium of thisgame in which each player uses the same strategy (i.e., a symmetric Bayesian Nashequilibrium).


Exercises

Exercises

I The market of lemons: Consider a seller of a used car and a potential buyer of thatcar. Suppose that quality of the car, θ, is a uniform draw from [0, 1]. This qualityis known to the seller, but not to the buyer. Suppose that the buyer can make anoffer p ∈ [0, 1] to the seller, and the seller can then decide whether to accept orreject the buyer’s offer. Payoffs are as follows:

uS =

pθ

if offer acceptedif offer rejected

uB =

a + bθ − p0

if offer acceptedif offer rejected

Assume that a ∈ [0, 1), that b ∈ (0, 2) , and that a + b > 1. These assumptionsimply that for all θ, it is more efficient for the buyer to own the car.

I Find the unique PSBNE of this game.


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Games With Incomplete Information: Theory of Auctions And ...hrtdmrt2/Teaching/GT_2017_19/L11.pdfGames With Incomplete Information: Theory of Auctions And Other Examples Carlos Hurtado