Download - If you are planning to buy a used car
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