Content map: SMU H3 Game Theory Map

Chapter 15: Auctions, Bidding Strategy and Auction Design

Continuous Random Variables

Concept

A continuous random variable takes values in a subset of the real line, such as $[0,1]$ .
A single exact value has probability $0$ .
Probabilities are described using events such as $X\leq x$ .

Definition

Definition:

The cumulative distribution function of $X$ is

F(x)=P(X\leq x)

For a random variable on $[0,1]$ , $F(0)=0$ , $F(1)=1$ , and $F$ is increasing.

Density

Definition:

If $F$ is differentiable, its density is

f(x)=F'(x)

Equivalently,

dF(x)=f(x)\,dx

Insight:

For continuous variables, probability comes from intervals, not points.

Expected Value

Key Formula

Definition:

If $X$ is distributed on $[0,1]$ with density $f$ , then

E(X)=\int_0^1 x f(x)\,dx

Using $dF(x)=f(x)\,dx$ ,

E(X)=\int_0^1 x\,dF(x)

Integration by Parts

Since $g(x)=x$ and $g'(x)=1$ ,

\int_0^1 x\,dF(x) = xF(x)\bigg|_0^1-\int_0^1 F(x)\,dx

With $F(1)=1$ and $F(0)=0$ ,

E(X)=1-\int_0^1 F(x)\,dx

Result:

For a continuous variable on $[0,1]$ ,

E(X)=\int_0^1 x f(x)\,dx =1-\int_0^1 F(x)\,dx

Uniform Distribution

Definition

Definition:

A uniform distribution assigns equal density to every point in an interval.

Uniform Distribution on $[0,1]$

The cumulative distribution function is

F(x)=x

The density is

f(x)=1

The expected value is

E(X)=\int_0^1 x\,dx=\frac{1}{2}

Uniform Distribution on $[a,b]$

The cumulative distribution function is

F(x)=\frac{x-a}{b-a}

The density is

f(x)=\frac{1}{b-a}

The expected value is

E(X)=\int_a^b x\frac{1}{b-a}\,dx=\frac{a+b}{2}

Insight:

For a uniform distribution, the expected value is the midpoint of the interval.

Conditional Expectation

Definition:

If $X$ has cumulative density $F(X)$ and density $f(X)$ on $[a,b]$ , then

E(X\mid X\leq v) =\frac{1}{F(v)}\int_a^v x f(x)\,dx =\frac{1}{F(v)}\int_a^v x\,dF(x)

Mixed Strategies as Distribution Functions

Concept

A continuous mixed strategy can be represented by a cumulative distribution function $F$ .
$F(x)$ is the probability that a randomly chosen bid is at most $x$ .

Independent Mixing

Definition:

If each of $n-1$ opponents independently uses the same mixed strategy $F$ , then the probability all opponent bids are at most $x$ is

F(x)^{n-1}

Equilibrium Logic

In a mixed-strategy equilibrium, every bid used with positive probability gives the same expected payoff.
This indifference condition pins down the distribution $F$ .
The support endpoints are found from $F(a)=0$ and $F(b)=1$ .

Insight:

Continuous mixed strategies are solved by making the player indifferent across every bid in the support.

What Are Auctions?

More Than Just Buying and Selling

Definition:

An auction is a mechanism for allocating scarce objects, which specifies who wins and how payments are determined. Auctions are useful when the seller does not know bidders’ valuations, as bidding behaviour reveals information about willingness to pay.

Auction Formats

Ascending-price auction: Price rises until only one bidder remains; also called an English auction.
Descending-price auction: Price starts high and falls until one bidder accepts; also called a Dutch auction.
First-price auction: Highest bidder wins and pays own bid.
Second-price auction: Highest bidder wins and pays the second-highest bid.
All-pay auction: All bidders pay, but only the highest bidder wins.
War of attrition: Players pay delay or effort costs until one player exits.

Information in Auctions

Independent Private Values: each bidder knows own value and other bidders’ values are independent draws.
Common Value: the object has the same ex-post value for all bidders, but bidders may have different information about that value.

Strategic Elements

Asymmetries of information can arise between the seller and bidders, and also among bidders.
These information asymmetries create screening and signalling incentives.
Optimal strategies depend on risk attitudes, how values are determined, and the auction format.
Expected revenue may or may not change with the auction format adopted.

Insight:

Auction design changes incentives by changing what the bid controls: winning probability, payment conditional on winning, information revelation, or costly persistence.

The Winner’s Curse

Concept

The winner’s curse arises in common-value auctions.
Winning is informative: it often means the winner had the most optimistic signal.
A bidder who ignores this selection effect tends to overpay.

Reasoning

Let $V$ be the common value and $s_i$ be bidder $i$ ‘s signal.
The relevant value after winning is conditional:

E(V\mid s_i,\text{ win})

Usually,

E(V\mid s_i,\text{ win})<E(V\mid s_i)

Bids must be shaded to account for the bad news contained in winning.

Result:

In common-value auctions, rational bidders condition on winning before deciding how much to bid.

Bidding in Auctions

First-Price Auction

The winner pays his own bid.
A bidder shades below own value.
To find the optimal bid, consider a bidder with value $v_1$ who bids $b$ .
The bidder wins if all opponents have values below $b$ which occurs with probability $F(b)^{n-1}$ .
Expected payoff conditional on winning is $(v_1 - b) \cdot F(b)^{n-1}$ .
Maximising over $b$ yields the equilibrium bid function.

Result:

With value $v_1$ , the bid equals the expected largest opponent value conditional on all opponents having values below $v_1$ :

E\left[\max\{v_2,\dots,v_n\}\mid v_1>\max\{v_2,\dots,v_n\}\right]

First-Price Common-Value Auction with Independent Signals

The object has one common value $V$ .
Each bidder observes an independent signal $s_i$ about $V$ .
With a monotone symmetric bid function, bidding as type $z$ wins when all opponent signals are below $z$ .
Expected payoff from bidding as type $z$ is

\left(E(V\mid s_i,\,Y<z)-b(z)\right)P(Y<z)

where $Y=\max\{s_j:j\neq i\}$ .

In equilibrium, set $z=s_i$ and shade the bid below the conditional value after accounting for winning.

Result:

In a first-price common-value auction, the bid is based on

E(V\mid s_i,\,Y<s_i)

not on the unconditional estimate $E(V\mid s_i)$ .

Second-Price Auction

This is a second-price auction.
The winner pays the second-highest bid.

Result:

With independent private values, truthful bidding is weakly dominant:

b(v)=v

Second-Price Common-Value Auction with Independent Signals

Truthful bidding is no longer dominant because the value is common, not private.
Winning at a given bid means the bidder’s signal is high relative to others.
If the second-highest signal is $Y$ , the pivotal event is $Y=s_i$ .
The equilibrium bid equals the value estimate conditional on just winning:

b(s_i)=E(V\mid s_i,\,Y=s_i)

Result:

In a second-price common-value auction, bid the expected common value conditional on being exactly pivotal:

b(s_i)=E(V\mid s_i,\,Y=s_i)

All-Pay Auction

Everyone pays their bid.
This is a first-price all-pay auction.
Only the highest bidder receives the prize.
No pure-strategy equilibrium exists in the standard continuous version.

Result:

For prize value $1$ , the symmetric mixed-strategy equilibrium for $n$ bidders has support $[0,1]$ and

F(x)=x^{1/(n-1)}

War of Attrition

This is a second-price all-pay auction.
Players choose how long to remain in the contest.
Remaining is costly, and the player who stays longer wins.

Result:

If $\alpha_i=\frac{1}{v_i}$ , the two-player equilibrium CDFs are

F_1(x)=1-e^{-\alpha_2 x}, \qquad F_2(x)=1-e^{-\alpha_1 x}

Game Theory Chapter 15: Auctions, Bidding Strategy and Auction Design

Chapter 15: Auctions, Bidding Strategy and Auction Design

Continuous Random Variables

Concept

Definition

Density

Expected Value

Key Formula

Integration by Parts

Uniform Distribution

Definition

Uniform Distribution on [0,1][0,1][0,1]

Uniform Distribution on [a,b][a,b][a,b]

Conditional Expectation

Conditional Expectation

Mixed Strategies as Distribution Functions

Concept

Independent Mixing

Equilibrium Logic

What Are Auctions?

More Than Just Buying and Selling

Auction Formats

Information in Auctions

Strategic Elements

The Winner’s Curse

Concept

Reasoning

Bidding in Auctions

First-Price Auction

First-Price Common-Value Auction with Independent Signals

Second-Price Auction

Second-Price Common-Value Auction with Independent Signals

All-Pay Auction

War of Attrition

Uniform Distribution on $[0,1]$

Uniform Distribution on $[a,b]$