�ݺ�ߣ

GAME THEORY
Easy College Study

Lecture 5: Bertrand Duopoly, Crime
Report and Auctions
■ In this lecture, we are going to see applications of static game theoretical model in real
world, to understand production decision making, moral dilemma, and auction process
■ Review: definition of games, strategies, NE and Cournot Duopoly
■ Bertrand Duopoly: continuous and discrete pricing
■ Reporting a crime:
– How can we understand the inefficiency in social moral dilemmas like this ?
■ Auctions
– Complete information
– Second-price sealed-bid auction
– First-price sealed-bid auction
– Spectrum Auctions: some facts

Review: Best response
■ The best response captures the best choice of individual i when given other players
actions a-i

Review: Pure strategy Nash Equilibrium
(PNE)

Review: Mixed strategy
■ Denote ∆ 𝐴𝑖 = {𝛼𝑖}, then ∆ 𝐴𝑖 is a (|𝐴𝑖| − 1)-simplex
■ Denote ∆ 𝐴 the set of lottery over A, then ∆ 𝐴 is a (|A| − 1)-simplex
■ Then, each prole of mixed strategies 𝛼 = (𝛼1, … , 𝛼 𝑛), maps to a lottery on ∆ 𝐴 ,
with assumption of independence, we have

Review: Mixed strategy Nash Equilibrium

Cournot Duopoly
■ Two firms i and j, compete by producing quantities 𝑞𝑖 ≥ 0, and 𝑞 𝑗 ≥ 0. under price
function 𝑃 𝑞𝑖, 𝑞 𝑗 = 𝑎 − 𝑏(𝑞𝑖 + 𝑞 𝑗), 𝑎, 𝑏 > 0. Marginal cost of production c>0
■ Each firm maximizes profit given the other's production. For firm i :
■ First-order condition: 𝑎 − 2𝑏𝑞𝑖 − 𝑏𝑞 𝑗 − 𝑐 = 0
■ Solving for 𝑞𝑖 gives firm i's best response:
■ Likewise, we can derive firm j’s best response

Bertrand Duopoly
■ While Cournot Duopoly is competing in quantity, Bertrand Duopoly, however, is
competing in pricing strategy
■ Two firms i and j compete by setting prices 𝑝𝑖 ≥ 0 and 𝑝𝑗 ≥ 0, resp. Consumers buy
only the cheapest good; firm i faces demand function:

Bertrand Duopoly
■ Each firm maximizes profit 𝜋 given the other's price. For firm i :

Bertrand Duopoly: discrete pricing
■ Assume firms can set prices equal to some multiple of the smallest denomination
cent (e.g., 1 penny USD)?
■ Refine 𝑝𝑖
𝑚
to maximize 𝜋(𝑝𝑖, 𝑝𝑗) when 𝑝𝑗 ≫ 0 subject to 𝑝𝑖
𝑚
dividable by cents.

Bertrand Duopoly: discrete pricing

Reporting a crime
■ Players: n > 1 bystanders.
■ Actions: Ai = {(R)eport, (D)on’t Report}.
■ Pareto efficient outcome: exactly one bystander reports.
■ Best response:
■ Report if no one else reports, Don't Report if anyone else reports.
■ Is existence of symmetric MNE guaranteed?

Reporting a crime
■ Symmetric MNE: p = 1 - (c/v )1/(n-1) for each i ∈ N.
■ Probability i reports, p, is decreasing in n: increasing incentives to “free load" as
more bystanders witness the crime (as n ↑).
■ Probability no one reports, (1 - p)n = (c/v )n/(n-1), is also decreasing in n(!): as more
bystanders witness the crime, the “public good" of reporting the crime is more under
provided.
■ Equilibrium welfare? What inefficiencies are there?

Auctions: complete information
■ In this class: we dene an “auction" as a market mechanism in which a single good is
sold to a player (“bidder") who submits a highest bid.
■ More generally: auctions may include multiple units, and more complicated actions
than a single bid, e.g., “bidding fee" auctions, and “Simultaneous Multiple-Round"
(SMR) auctions (more later).

Auctions: complete information
■ Types of auctions (we study):
■ Second-price sealed-bid auction (SPA): static (simultaneous move) auctions where
highest bidder pays the second highest bid. SPA approximates dynamic assenting price
“English" auction.
■ First-price sealed-bid auction (FPA): static (simultaneous move) auctions where highest
bidder pays their bid. FPA approximates dynamic descending price “Dutch" auction.
■ Auctions can either be under complete information (covered today), or
incomplete/private information
■ Both SPA and FPA under complete information yield multiple PNE...

Auctions: Second-price sealed-bid auction (SPA)
■ Players: n > 1 bidders.
■ Actions: bids bi ≥ 0 for each i ∈ N.

Auctions: Second-price sealed-bid auction (SPA)

Auctions: Second-price sealed-bid auction
(SPA)

Auctions: First-price sealed-bid auction
(FPA)
■ Players: n > 1 bidders.
■ Actions: bids bi ≥ 0 for each i ∈ N.

Auctions: First-price sealed-bid auction (FPA)

Auctions: First-price sealed-bid auction
(FPA)

�ݺ�ߣ

L5

More Related Content

L5