Keisuke Kawata
ISS, UTokyo
Search decision making
Sequential search: → Agent sequentially samples (learns) the private value of alternative.
Why sequential search?： Sequential search is the simplest model of search friction ←can focus to understand the search decision making.
Key concepts: Dynamic optimization problem with uncertainly, threshold strategies
Contents
1. Basic environments
2. Optimal strategy in finite alternatives 3. Optimal strategy in infinite alternatives
1. Basic environment
• A age t e.g., o su e ; jo seeke ; fi a se ue tially lea the alue (e.g., individual consumer surplus; wage) of alternatives (e.g., goods; job; best worker; supply source; business solutions and investment).
← “e ue tial sa pli g of alte ati es, o dy a i lotte y .
• Strategy: stop the search activities or continue.
• Alte ati e’s alue is d a f o a i depe de t a d ide ti ally dist i uted iid) of with support [ , ∞ . ←Common knowledge.
1. Classification
• Conventional sequential search models can be categorized by two attributes. Number of alternatives: Finite/Infinite number.
Recalling: Agent can/cannot recall the value of previous alternatives and freely come back.
• The slide will discuss the sequential search without recall
Finite alternatives Infinite alternatives Full memory Consumer search Consumer search
No memory Not tractable Job search
2. Finite alternatives
• Timing of game: In each period t
1. Observing the value of an alternative (denoted by ).
2. Decision-making whether to accept the alternative/ continue to search.
→ Stop: Obtaining payoff (becoming employees)
→ Continue: Moving next alternative (period) with search cost c.
• We here discuss a case with three alternatives.
• For simplicity, zero payoff if no alternatives are accepted.
2. Optimal strategy
• Applying the backward induction method←Start from period 3.
Period 3: Because the agent cannot recall jobs 1 and 2, the job 3 must be accepted, and her payoff is then .
Period 2: After observing (but not observing ), she decides whether to stop or continue to search. ← Her strategy is mapping from to stop/continue to search.
2. Optimal strategy (cont.)
• If the alternative 2 is accepted, payoff is , while the payoff to continue the job- search is
∫ − .
• Continuing to search ↔
∫ − ≥
↔∫ − ≥ (Stopping condition)
• Because the right-hand side of the stopping condition is decreasing in , the condition can be rewritten by the threshold strategy as
� � � ℎ ↔ < ҧ
where
∫ − ҧ = .
2. Optimal strategy (cont.)
• Note that expected payoffs at the decision-making are
= and
= max , ∫ −
= + max , ∫ − −
→ Value of alternative 2 plus the option value of search.
If = , the expected payoff at period 2 must be larger than at period 3 due to the option value.
2. Optimal strategy (cont.)
• Characterizing the optimal threshold strategy in period 1.
• The payoff accepting the alternative 1 is , while the payoff continuing to search is
∫ − .
The search threshold is obtained by
∫ − = ҧ
which can be rewritten as
∫ − ҧ + max , ∫ − − =
• Age t is o e pi ky i pe iod 1 ҧ ≥ ҧ ) due to the option value
← ҧ = ҧ if the option value is zero.
2. Main finding
• The discussion can be extend more general cases; the number of alternative is more than 3 (but still finite number), and/or risk averse agent.
→ The job-acceptance (stopping) probability is generally increasing according to period because the option value is decreasing. (Experimental Evidence)
Intuition; In earlier periods, even if drawing value is lower, an agent has a change to recover by continuing search activity.
• Main reason of less tractability.
2. Empirical/Experimental test
• The model predicts the acceptance probabilities are increasing.
• Experimental and/or observable data (e.g., unemployment, lab-experiments) allows us to observe the acceptance probabilities among group.
← In same case, the acceptance probabilities are increasing over time period
• Inconsistent with the model prediction?
←Survival bias (Agents with lower threshold tend to quit from the observation poor in earlier periods).
• In experiment, the strategic design is better?
3. Infinite alternatives
• Characterizing the infinite number of alternatives.
• Now considering a general case with the dynamic structure.
• The life-time utility is
=
� +
where + = if the agent have accepted a alternative with wage v, while + =
− if the agent have not accepted any alternatives.
← Assuming no financial markets.
3. Value function (cont)
• Applying the DP technique.
• Choice variable: whether to accept an alternative v or not
→ An indicator choice variable I(v): v → {0,1}.
• Assuming v is supported with [ , ].
• The life-time utility (value) before observing a alternative, , is
= max න � I + − � − + � +
= න �max , − + � +
3. Value function (cont)
• The optimal search strategy can still be characterized by the threshold strategy Continue to search iff ≥ ҧ where
ҧ = − + � + .
• In cases with the infinite number of alternatives, the life-time utility and then threshold is time invariant because of the constant option value.
→ Search threshold is
ҧ = − + �
where = = +
3. Optimal threshold
• With threshold, value function can be rewritten as
= නത
� + ҧ − + � ,
and then
= ∫ത
� − ҧ
− � ҧ
3. Property of optimal threshold
• The threshold can be obtained as
− � ҧ = − + � න
ത
� − ҧ
• Only if � → and c → , ҧ → , (accept only alternatives with )
←Converging Walrasian (Frictionless) market.
• Intuition: The dynamic sequential search model includes not only the explicit search costs c but also implicit time costs ←disappeared if no-time discounting.
3. Property of optimal threshold
• We can obtain the same strategy by solving
= maxത න
ത
� + ҧ − + � ,
3. Comparative statistics
• Because the optimal threshold is obtained as implicit form, we need to use the total differentiation.
Example) Let obtain the comparative statistics result over c
Assignment
• Incorporating additional dynamic structure.
• If an agent has accepted an alternative with v, her value is
= + � + − ,
while the value of searching is
= max න � I + − � − + � +
Assignment: Characterizing an optimal threshold value.
4. Application: Collective Decision-making
• In some cases (e.g., search for new project, best worker, and supply source), the search decision-making is by multiple agents (e.g., members of operating
committee).
→ Albrecht; Anderson; Vroman (2010) extends the sequential search model into the committee decision-making by voting.
• Committee decides whether to accept a current alternative or reject and continue search activity.
• Values of the accepted alternative are different among committee members
←Private value is drawn from the iid distribution, F v , where is the private
4. Application: Collective Decision-making
• Committee is characterized by {N,M}, where N is the number of members, and M is the number of votes required to accept an alternative.
→McCall model is a special case as N=M=1. Timing: In each period,
1. Each member observes her private value of an alternative �. 2. Voting whether to accept the alternative/ continue to search. Number of votes to accept is,
→ larger than M: Obtaining payoff is �
→ smaller than M: Moving next alternative with search cost c.
4. Application: Collective Decision-making
• Focusing on stationary Markov strategies; individual voting strategy still depends on only the private value of current alternative, ��: � → , .
• Voting strategy is still characterized as the threshold strategy ҧ�.
→ Member c votes to accept ↔ � ≥ ҧ�
Definition: Committee search equilibrium
• Equilibrium voting threshold ҧ� maximizes life-time utility of member c.
4. Application: Collective Decision-making
• Let focus on a simple example case as N=M=2, and the committee is then consisted by member 1 and 2.
• Considering the strategy of member 1. Because M=2, if < ҧ , the alternative is never accepted. If ≥ ҧ , the alternative is accepted if and only if member 1 votes to accept (pivotal voter).
• The value functions are
= maxത
1 ҧ − + �
+ − ҧ ҧ − + � + න
ത1
�
4. Application: Collective Decision-making
• First order condition is
ҧ = − + � .
← Same form in the individual decision-making case (Pivotal voting). However, is smaller than the individual decision-making case as
= − ҧ + − ҧ ҧ + − ҧ ∫ത1
�
− ҧ � − − ҧ ҧ �
← Smaller than the individual case (less picky) if ҧ > .
4. Application: Collective Decision-making
• Symmetric equilibrium ( ҧ = ҧ = ҧ) holds.
− � ҧ = − + � − ҧ න
ത
� − ҧ
• Individual trade-off is same as in the individual decision-making case.
→ But the value to continue the search activity (U) is decreased because any members cannot perfectly control the committee decision-making in the future voting.
→ Less picky.
Experimental evidence: Hizen; Kawata; Sasaki (15)
Conclusion
• We discuss about the fundamental problem of search decision-making in sequential search situation.
→ Optimal strategy can be characterized by the threshold strategy.
• McCall model can explain the dispersion of individual payoff (violation of the law of one price) even as a result of optimal behavior.
→If search costs is larger, the individual payoff is more dispersed. Shortcoming: Exogeneous distribution of alternative value
Seminal paper
McCall, J. J. (1970). Economics of information and job search. Quarterly Journal of Economics, 113-126.
Committee search
Albrecht, J., Anderson, A., & Vroman, S. (2010). Search by committee. Journal of Economic Theory, 145(4), 1386-1407.
Compte, O., & Jehiel, P. (2010). Bargaining and majority rules: A collective search perspective. Journal of Political Economy, 118(2), 189-221.
Delegated search
Lewis, T. R. (2012). A theory of delegated search for the best alternative. RAND Journal of Economics, 43(3), 391-416.
Ulbricht, R. (2016). Optimal delegated search with adverse selection and moral hazard. Theoretical Economics, 11(1), 253-278.
