Optimal stopping, satisficing and scarce. attention. Pantelis Pipergias Analytis. attention. 1 Cornell University.

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1 1 Cornell University January 29, 2018

2 The secretary problem

3 Assumptions Single position to fill. There are n applicants for the position, and the value of n is known. The applicants, if seen altogether, can be ranked from best to worst unambiguously. The applicants are interviewed sequentially in random order, with each order being equally likely. Immediately after an interview, the interviewed applicant is either accepted or rejected. The decision to accept or reject an applicant can be based only on the relative ranks of the applicants. The objective of the general solution is to have the highest probability of selecting the best applicant.

4 Performance of different heuristic strategies

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6 Searching for a book

7 The random search world (sequential search) 6 utility Utility Index

8 The random search model R n (u 1, u 2,..., u n ) = max(u 1, u 2,...u n ) nc (1) R(u n, c) = P (u n > max(u 1, u 2,...u n 1 )) (2) E(u n max(u 1, u 2,...u n 1 ) u n > max(u 1, u 2,...u n 1 )) c stopping rule Sample the alternatives at random if u n > T opt (c), stop sampling (3) if u n T opt (c), continue sampling

9 Discussion: Online dating as a search problem

10 The Microeconomics 101 world Utility 5.0 utility George's opinion

11 The discrete choice world 10 8 Utility 6 utility Record's av. review

12 A simple model R(A k ) = P (u k > y) E(u k y c u k > y) P (u k y) c = P (u k > y) E(u k y u k > y) c. (4) If for two alternatives A i, A j S, f(a i ) > f(a j ) then R(A i ) > R(A j ). Selection rule: Order the alternatives based on their unconditional expectation. Select the items for sampling in this order. Stopping rule: If at any stage subjective expected gain is negative, terminate the search.

13 A concrete example Utility EU RU K L M 0.92 NA M^ L K Average rating P (u L > u K ) = 0.33, E(u L u K u L > u K ) = P (u M > u L ) = and E(u M u L u M > u L ) = 0.152

14 Multi-attribute utility models Multi-linear utility (MLU): f(a i1,.., a im ) = j β ja ij It is one of the cornerstone models in multi-attribute decision making (Keeney and Raiffa, 1993). Equal weighted linear utility (EW): f(a i1,.., a im ) = j a i j, where all the attributes a i1,.., a im are normalized and brought on the same scale. Originally, suggested by Dawes and Corrigan (1974). Single attribute utility (SA): f(a i1,.., a im ) = a ij where a ij has highest ecological validity amongst {a i1,.., a im } as it is expressed by Kendall s tau non-parametric correlation. Initially studied by Hogarth and Kareleia (2005).

15 Discussion: Online marketplaces and search

16 The eyetracking study: design

17 The eyetracking study: results

18 The eyetracking study: design Percentage % of fixations % of clicks Rank of Abstract Figure 1: Percentage of times an abstract was viewed/clicked depending on the rank of the result.

19 Click models ur 1q urq Ar 1 document u r 1 Ar document u r Cr 1 Cr... Er 1 Er... Figure 3.2: Graphical representation of the cascade model (fragment).

20 Bestseller search: an example

21 Bestseller search: another example

22 Bestseller search: yet another example

23 The music lab experiment (Salganik, Watts and Dodds, 2006)

24 Inequality in markets for cultural products

25 Unpredictability in markets for cultural products

26 Introducing social influence and variability in taste The market is populated by M agents A 1,..., A M. Objective utility component u o identical for all the agents, and subjective component u s, agent specific. The overall utility u i = u oi + u si. The objective component and subjective component are (iid) random variable with mean µ and variance σ 2. Overall variance σ 2 = σ 2 oi + σ2 si. Popularity heuristic The agents sample the alternatives X 1, X 2...X n in decreasing order of popularity P 1 > P 2 >... > P n, if u n > T pop, stop sampling (5) if u n T pop, continue sampling

27 Popularity search: an example 7.5 utility 10.0 Utility Popularity Note that for c = 0 the two processes imply a perfect knowledge market.

28 Discussion: Ranking in social media

29 Probability of choosing a sampled alternative If the alternative i is searched, it is accepted with some probability p i. p i = P r(u i,s + u i,o > T ) = P r(u i,s > T u i,o ). In a realization of the environment, the threshold T and the u i,o s are fixed, while u i,s is a random variable

30 Probability of choosing an alternative Suppose the agent has decided to search the alternatives with search order δ. The probability that the agent searches and accepts the alternative δ 2 is (1 p δ1 ) p δ2. Similarly, the probability that the agent searches and accepts the alternative δ 3 is (1 p δ1 ) (1 p δ2 ) p δ3. In general, the probability that the agent searches and accepts the alternative δ k is k 1 (1 p δi ) p δk. i=1

31 Ordering We define its ordering as a pair of vectors (γ, ζ) that tell us which alternative has the highest popularity, which one is second, etc., taking into account the possible ties. Definition For a vector P, we call the pair (γ, ζ) its ordering. If ζ consists only of 0 s, that is if all popularities are different, then we call (γ, ζ) a strict ordering.

32 ordering Definition We call an ordering (γ, ζ) optimal if it maximizes the agents expected return. Suppose that 0 < p i < 1 for all i. Let f i denote the probability density function of u i and define g i to be the expected value of u i conditioned on u i > T. That is, g i = T u if i (u i )du i T f i(u i )du i. Then (γ, ζ) maximizes the expected utility for an agent if and only if g γ1... g γn and g γi = g γi+1 whenever ζ i = 1.

33 Terminal orderings and settling Definition Let r n be the ordering of the popularity vector at time n. We say that r n settles on (γ, ζ) R, if there exists some n 0, such that r n = (γ, ζ), for all n n 0. We say that an ordering (γ, ζ) R is terminal, if P r (r n settles on (γ, ζ)) > 0. Otherwise, we say that (γ, ζ) is non-terminal. Theorem The ordering of the popularity vector r n eventually settles on some ordering with probability 1. If 0 < p i < 1 for each i, then every non-strict ordering (γ, ζ) is non-terminal. Therefore, r n will settle on some strict ordering.

34 Strict ordering and existence proof Part of Theorem 4 states that non-strict orderings are always non-terminal, except in trivial cases. But a strict ordering might be either terminal or non-terminal. The following Theorem gives necessary and sufficient conditions for a strict ordering to be terminal. Theorem A strict ordering γ is terminal if and only if q γ γ 1 > q γ γ 2 >... > q γ γ N.

35 Eventual market shares Corollary A strict ordering γ with the property p γ1... p γn is terminal. Theorem Suppose that r n settles on the strict ordering γ. Then, for each i, Pn i lim n n = qγ i, a.s.

36 An example Imagine the market for textbooks in Advanced Microeconomics. Three alternatives with objective qualities u 1 = 1.525, u 2 = 1 and u 3 = subjective utilities u i,s are Gaussian random variables with mean 0 and a given variance σ 2 = 1 and the threshold T is equal to 1. The above objective utilities have been chosen so that, approximately, p 1 = 0.7, p 2 = 0.5 and p 3 = 0.3 ordering p 1 > p 2 > p 3. cost = 0.5

37 An example γ q 1 q 2 q 3 Terminal ū ˆp γ 1,2, Yes ,3, No ,1, Yes ,3, No ,2, No ,1, No

38 Average welfare Average utility and cost c = Gross utility Net utility Cost Diversity of preferences

39 Welfare dynamics Net utility d = 0 d = 0.2 d = 0.5 d = 0.8 Total cost d = 1 d = d = d = 0 d = 0.2 d = Block

40 End state distributions

41 Market inequality 1.0 Popularity search Random search 0.9 Gini coefficient d = 0.5 d = Search cost

42 Market unpredictability 1.00 Popularity search d = 0.5 Random search d = 0.5 Market unpredictability coefficient Search cost

43 Summary New model of popularity dynamics. Simple mechanism that explains the inequality and unpredictability....and stresses the importance of quality and taste differences. It generates predictions for the collection of empirical data in the future. The model illustrates why some heterogeneity in taste can be beneficial for collectives.