IF YOU ARE PLANNING TO BUY A USED CAR

Logon to

Examine

Buy

End

Start

No

Yes

HOW TO DECIDE: IT COULD BE A LEMON

Earlier people would take it to a mechanic for a careful examination

OVERVIEW

• The model• Characterizing optimal strategies for• the searcher• the expert•Market design

SEQUENTIAL MODEL

The SearcherSearch Cost:

Noisy signalReject & Start

again

Buy & Terminate

Query an Expert for fee

THE MODEL

• Expert and Searcher: Rational Agents• Expert sets profit maximizing fee• Searcher formulates utility maximizing

strategy• Stackelberg game

8

STACKELBERG GAMES

• Model• One player (leader) has dominate influence over another• Typically there are two stages• One player moves first• Then the other follows in the second stage• Can be generalized to have

• multiple groups of players• Static games in both stages

• Main Theme• Leader plays by backwards induction, based on the

anticipated behavior of his/her follower.

NOTATION

• = Search cost, • = Expert’s fee per query• : distribution of signals• : conditional distribution of values given signal• : Expected value of Searcher’s Utility • : Expected value of Expert’s profit

DISTRIBUTIONS

fv(x)

fs(y|v)

fv(y|s)

fs(x)

𝑃 (𝑣|𝑠 )=𝑃 (𝑠|𝑣 ) 𝑃 (𝑣 )

𝑃 (𝑠)

OVERVIEW

• The model•Characterizing optimal strategies for• the searcher • the expert•Market design

IF HAVING NO EXPERT…

•… but rather just noisy signals:• S – signals for which we buy

HSGN

• restriction that higher signal values are “good news" in the sense that when s1 > s2, the conditional distribution of v given s1 first-order stochastically dominates that of v given s2

(Wright 1986, Milgrom 1981): • if , then,

• The condition requires that the probability that the actual value is greater than any particular value v is greater for the case where the searcher received signal s1.

MAIN CLAIM

• The proof is based on showing that, if according to the optimal search strategy the searcher should resume her search given a signal s, then she must necessarily also do so given any other signal s’ < s• Use V to denote the expected value of

continuing the search

MAIN CLAIM (CONT.)

• If better to resume search given s:• Given HSGN: • Therefore:

• Now:

• Optimize according to t to get part (a) of the theorem

NOW WITH THE EXPERT…

•General Strategies:• Reject some (without consulting the

expert)• Accept some (without consulting the

expert)• Consult the expert and then decide if to

reject or accept

THEOREM

• Claim – given HSGN, the optimal strategy for the searcher is based on (tl,tu,v)

PROOF

• First show that if, optimally, the searcher should resume her search given a signal s, then she must also do so given any signal s’ < s• If the optimal strategy given signal s is to

resume search then the following two inequalities hold:

PROOF (2)

And similarly, if better to terminate for s then so is for s’>s

Based on HSGN

Pr(𝑠<

𝑡 𝑙)

Pr (𝑠>𝑡𝑢)Pr (𝑡

𝑙 ≤ 𝑠≤ 𝑡𝑢 )

Pr(𝑣<𝑉

𝑐 𝑒)

Pr (𝑣>𝑉𝑐𝑒 )

BOTTOM LINE

• For satisfying the HSGN assumption, the optimal search strategy can be described by the tuple • where: • : search should be resumed; • search should be terminated • : Query the expert and accept and terminate if

the value obtained from the expert is above the expected value of resuming the search,, otherwise search should resume

EXPERT’S PROFIT

c¤e :2= 2

CALCULATING

the expected overall number of queries made

AN EXAMPLE

s

¿

EFFECT OF cs ON SIGNAL THRESHOLD KEEPING CONSTANT

Reject

Accept

Query & decide

EFFECT OF ce AND de

OVERVIEW

• The model• Characterizing optimal strategies• the searcher• the expert•Market design

SOCIAL WELFARE

• Social welfare = Searcher’s Utility + Expert’s

profit

• Expert’s Profit :• Searcher’s Utility :

MARKET DESIGN

•Does it make sense for market designer to subsidize the query cost of an expert

MARKET DESIGN

• pays lump-sum amount b to the expert to reduce the query cost from to and new social welfare W’

EFFECT OF ON SUBSIDY

• At social welfare maximizing subsidy:

• Digital services with close to zero marginal cost should be provided for free.

CLAIM

• The optimal level of subsidy for an additive measure of social utility is the level that forces the buyer to exactly fully internalize the cost of provision of expert services