The complexity of shelflisting

Abstract

Optimal shelflisting invites profit maximization to become sensitive to the ways in which purchasing decisions are order-dependent. We study the computational complexity of the corresponding product arrangement problem when consumers are either rational maximizers, use a satisficing procedure, or apply successive choice. The complexity results we report are shown to crucially depend on the size of the top cycle in consumers’ preferences over products and on the direction in which alternatives on the shelf are encountered.

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Notes

  1. 1.

    The Restricted Betweenness problem is not explicitly studied in Opatrny (1979). However, in the NP-hardness reduction for the Betweenness problem in Opatrny (1979), the instance constructed is an instance of the Restricted Betweenness problem.

References

  1. Apesteguia, J., & Ballester, M. A. (2010). The computational complexity of rationalizing behavior. Journal of Mathematical Economics, 46(3), 356–363.

    Article  Google Scholar 

  2. Ausiello, G., Protasi, M., Marchetti-Spaccamela, A., Gambosi, G., Crescenzi, P., & Kann, V. (1999). Complexity and approximation: Combinatorial Optimization Problems and their approximability properties. Secaucus, NJ: Springer.

    Google Scholar 

  3. Bernheim, B. D., & Rangel, A. (2009). Beyond revealed preference: Choice-theoretic foundations for behavioral welfare economics. Quarterly Journal of Economics, 124(1), 51–104.

    Article  Google Scholar 

  4. Cormen, T. H., Leiserson, C. E., Rivest, R. L., & Stein, C. (2009). Introduction to algorithms (3rd ed.). Cambridge: MIT Press.

    Google Scholar 

  5. Eliaz, K., & Spiegler, R. (2011). Consideration sets and competitive marketing. Review of Economic Studies, 78(1), 235–262.

    Article  Google Scholar 

  6. Garey, M. R., & Johnson, D. S. (1979). Computers and intractability: A guide to the theory of NP-completeness. New York: W. H. Freeman.

    Google Scholar 

  7. Gigerenzer, G., Todd, P. M., & Group AR. (1999). Simple heuristics that make us smart. Oxford: Oxford University Press.

    Google Scholar 

  8. Leininger, W. (1993). The fatal vote: Berlin versus Bonn. FinanzArchiv, 50(1), 1–20.

    Google Scholar 

  9. Manrai, A. K., & Sinha, P. (1989). Elimination-by-cutoffs. Marketing Science, 8(2), 133–152.

    Article  Google Scholar 

  10. Masatlioglu, Y., & Ok, E. A. (2005). Rational choice with a status-quo bias. Journal of Economic Theory, 121(1), 1–29.

    Article  Google Scholar 

  11. Opatrny, J. (1979). Total ordering problem. SIAM Journal on Computing, 8(1), 111–114.

    Article  Google Scholar 

  12. Rubinstein, A., & Salant, Y. (2006). A model of choice from lists. Theoretical Economics, 1(1), 3–17.

    Google Scholar 

  13. Salant, Y. (2003). Limited computational resources favor rationality. Discussion Paper.

  14. Salant, Y. (2011). Procedural analysis of choice rules with applications to bounded rationality. American Economic Review, 101(2), 724–748.

    Article  Google Scholar 

  15. Salant, Y., & Rubinstein, A. (2008). (A, f): Choice with frames. Review of Economic Studies, 75(4), 1287–1296.

    Article  Google Scholar 

  16. Sharir, M. (1981). A strong-connectivity algorithm and its applications in data flow analysis. Computers & Mathematics with Applications, 7(1), 67–72.

    Article  Google Scholar 

  17. Simon, H. A. (1955). A behavioral model of rational choice. Quarterly Journal of Economics, 69(1), 99–118.

    Article  Google Scholar 

  18. Spiegler, R. (2014). Competitive framing. American Economic Journal: Microeconomics, 6(3), 35–58.

    Google Scholar 

  19. Tarjan, R. (1972). Depth-first search and linear graph algorithms. SIAM Journal on Computing, 1(2), 146–160.

    Article  Google Scholar 

  20. Tovey, C. A. (2002). Tutorial on computational complexity. Interfaces, 32(3), 30–61.

    Article  Google Scholar 

  21. Valenzuela, A., & Raghubir, P. (2009). Position-biased beliefs: The center-stage efect. Journal of Consumer Psychology, 19(2), 185–196.

    Article  Google Scholar 

  22. Valenzuela, A., Raghubir, P., & Mitakakis, C. (2013). Shelf space schemes: Myth or reality? Journal of Business Research, 66(7), 881–888.

    Article  Google Scholar 

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Acknowledgements

We are grateful to two anonymous referees for their helpful comments and suggestions.

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Correspondence to Dinko Dimitrov.

Appendices

Appendix A

This Appendix gives a linear-time algorithm for calculating the top cycle of tournament preferences used in Theorem 5.

figurea

Appendix B

This Appendix includes Figures and Tables used in Theorem 7. In all tables shown below, \((\{a,b\}, c, d)\) represents the lists (abcd) and (bacd).

Fig. 1
figure1

This figure shows the top favorite products and the preferences of the four buyers created for a \((u_i,v_j,u_k)\in \mathcal {C}\) in the NP-hardness of the SE-PA-\(f^{{\text {SC}}}\) problem in Theorem 7. An arc from a product to another product means that the former one is preferred to the latter one

Fig. 2
figure2

This figure shows the top favorite products and the preferences of the 14 buyers created for a \((v_i,w,u_j)\in \mathcal {D}\) in the NP-hardness of the SE-PA-\(f^{{\text {SC}}}\) problem in Theorem 7. The integer on the left side of each graph is the number of buyers with the top cycle and preferences as shown in the graph. An arc from a product to another product means that the former one is preferred to the latter one

Table 2 This table summarizes all lists of the top favorite products \(p(u_i)\), \(p(v_j)\), \(p(u_k)\), and \(d_1\) of the first two buyers corresponding to a 3-tuple \((u_i, v_j, u_k)\in \mathcal {C}\) (see the two graphs above in Fig. 1 for the preferences of these two buyers). The results are the products that the two buyers will choose, one for each
Table 3 This table summarizes all lists of the top favorite products \(p(u_i)\), \(p(v_j)\), \(p(u_k)\), and \(d_2\) of the last two buyers corresponding to a 3-tuple \((u_i,v_j,u_k)\in \mathcal {C}\) (see the two graphs below in Fig. 1 for the preferences of these two buyers). The results are the products that the two buyers will choose, one for each
Table 4 This table summarizes all lists of the top favorite products \(p(v_i)\), \(p(u_j)\), p(w), and \(d_1\) of the first two buyers corresponding to a 3-tuple \((v_i,w,u_j)\in \mathcal {D}\) (see the two graphs above in Fig. 2 for the preferences of these two buyers). The results are the products that the two buyers will choose
Table 5 This table summarizes all lists of the top favorite products \(p(v_i)\), \(p(u_j)\), p(w), and \(d_2\) of the last \(5+7=12\) buyers corresponding to a 3-tuple \((v_i,w,u_j)\in \mathcal {D}\) (see the two graphs below in Fig. 2 for the preferences of these 12 buyers). In the column for results, the two products \(p, p'\) shown in each row mean that the first 5 buyers choose p and the last 7 buyers choose \(p'\). So, for a list with \(p, p'\) in the column for results, the total profit of the products chosen by the 12 buyers is \(5\mu (p)+7\mu (p')\)

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Yang, Y., Dimitrov, D. The complexity of shelflisting. Theory Decis 86, 123–141 (2019). https://doi.org/10.1007/s11238-018-9675-7

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Keywords

  • Bounded rationality
  • Choice from lists
  • Computational complexity
  • Product arrangement
  • Top cycle