```{youtube} eYTGQCGpmXI

```

Anders Munk-Nielsen put his code on github here <https://github.com/GamEconCph/Lectures-2021/tree/main/Bayesian%20Games>

Anders Munk-Nielsen put his code [on GitHub](https://github.com/GamEconCph/Lectures-2021/tree/main/Bayesian%20Games).

Much of our Python code below is based on his.

This is a **Bayesian game**, a Nash equilibrium of which is called a **Bayesian Nash equilibrium**.

To complete the specification of the situation, we'll assume that prospective buyers' valuations are independently and indentically distributed according to a probability distribution that is known by all bidders.

To complete the specification of the situation, we'll assume that prospective buyers' valuations are independently and identically distributed according to a probability distribution that is known by all bidders.

Bidder optimally chooses to bid less than $v_i$.

Bidder optimally chooses to bid less than $v_i$.

### Characterization of FPSB Auction

A proof for this assertion is available at this Wikepedia page about Vicker auctions (https://en.wikipedia.org/wiki/Vickrey_auction)

A proof for this assertion is available at the [Wikepedia page](https://en.wikipedia.org/wiki/Vickrey_auction) about Vickery auctions

+++

**Protocols:** In a second-price sealed-bid (SPSB) auction, the winner pays the second-highest bid.

## Characterization of SPSB Auction.

## Characterization of SPSB Auction

In a SPSB auction bidders optimally choose to bid their values.

Formally, a dominant strategy profile in a SPSB auction with a single, indivisible item has each bidder bidding its value.

A proof is provided at this Wikepedia page about Vicker auctions (https://en.wikipedia.org/wiki/Vickrey_auction)

A proof is provided at [the Wikepedia

page](https://en.wikipedia.org/wiki/Vickrey_auction) about Vicker auctions

+++

We assume valuation $v_{i}$ of bidder $i$ is distributed $v_{i} \stackrel{\text{i.i.d.}}{\sim} U(0,1)$.

Under this assumption, we can analytically compute probabilitiy distributions of prices bid in both FPSB and SPSB.

Under this assumption, we can analytically compute probability distributions of prices bid in both FPSB and SPSB.

We'll simulate outcomes and, by using a law of large numbers verify, that the simulated outcomes agree with analytical ones.

We'll simulate outcomes and, by using a law of large numbers, verify that the simulated outcomes agree with analytical ones.

We can use our simulation to illustrate a **Revenue Equivalence Theorem** that asserts that on average first-price and second-price sealed bid auctions provide a seller the same revenue.

To read about the revenue equivalence theorem, see this Wikepdia page (https://en.wikipedia.org/wiki/Revenue_equivalence)

To read about the revenue equivalence theorem, see [this Wikepedia page](https://en.wikipedia.org/wiki/Revenue_equivalence)

+++

\end{aligned}

$$

and the PDF of $y$ is $\tilde{f}_{n-1}(y) = (n-1)y^{n-2}$.

and the PDF of $y_i$ is $\tilde{f}_{n-1}(y) = (n-1)y^{n-2}$.

Then bidder $i$'s optimal bid in a **FPSB** auction is:

Let $b(v_{i})$ be the optimal bid in a FPSB auction.

The revenue equivlance theorem tells us that a bidder agent with value $v_{i}$ on average receives the same **payment** in the two types of auction.

The revenue equivalence theorem tells us that a bidder agent with value $v_{i}$ on average receives the same **payment** in the two types of auction.

Consequently,

+++

In equations {eq}`eq:optbid1` and {eq}`eq:optbid1`, we displayed formulas for optimal bids in a symmetric Bayesian Nash Equilibrium of a

a FPSB auction"

In equations {eq}`eq:optbid1` and {eq}`eq:optbid1`, we displayed formulas for

optimal bids in a symmetric Bayesian Nash Equilibrium of a FPSB auction.

$$

\mathbf{E}[y_{i} | y_{i} < v_{i}]