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Evaluability of paired comparison data in stochastic paired comparison models: Necessary and sufficient condition

László Gyarmati, Csaba Mihálykó, Eva Orbán-Mihálykó, András Mihálykó

TL;DR

The paper tackles the problem of data evaluability in three-option stochastic paired comparisons by deriving a necessary and sufficient condition for the existence and uniqueness of the maximum likelihood estimator, applicable to THMM3 and Davidson's D3. It casts evaluability into a graph-theoretic framework using directed and bidirectional edges to capture better and equal decisions, and proves that conditions A-NS, B-NS, and C-NS are necessary and sufficient for a unique maximizer (Theorem FOT). The authors demonstrate that these NS conditions predict evaluable data more frequently than previously known sufficiency criteria via extensive simulations, especially with smaller datasets, and discuss the practical implications for parameter estimation and ranking. The work also highlights potential extensions to more-than-three-option models and provides rigorousAppendix-based proofs leveraging analysis and graph theory to substantiate the results.

Abstract

In this paper, paired comparison models with stochastic background are investigated. We focus on the models that allow three options for choice. We estimate all parameters, the strength of the objects and the boundaries of equal decision, by maximum likelihood method. The existence and uniqueness of the estimator are key issues of the evaluation. Although a necessary and sufficient condition for the general case of three options has not been known until now, there are some different sufficient conditions that are formulated in the literature. In this paper, we provide a necessary and sufficient condition for the existence of a maximum and the uniqueness of the argument that maximizes the value, i.e. for the evaluability of the data in models of these types. By computer simulation, we present the efficiency of the condition, comparing it to the previously known sufficient conditions.

Evaluability of paired comparison data in stochastic paired comparison models: Necessary and sufficient condition

TL;DR

The paper tackles the problem of data evaluability in three-option stochastic paired comparisons by deriving a necessary and sufficient condition for the existence and uniqueness of the maximum likelihood estimator, applicable to THMM3 and Davidson's D3. It casts evaluability into a graph-theoretic framework using directed and bidirectional edges to capture better and equal decisions, and proves that conditions A-NS, B-NS, and C-NS are necessary and sufficient for a unique maximizer (Theorem FOT). The authors demonstrate that these NS conditions predict evaluable data more frequently than previously known sufficiency criteria via extensive simulations, especially with smaller datasets, and discuss the practical implications for parameter estimation and ranking. The work also highlights potential extensions to more-than-three-option models and provides rigorousAppendix-based proofs leveraging analysis and graph theory to substantiate the results.

Abstract

In this paper, paired comparison models with stochastic background are investigated. We focus on the models that allow three options for choice. We estimate all parameters, the strength of the objects and the boundaries of equal decision, by maximum likelihood method. The existence and uniqueness of the estimator are key issues of the evaluation. Although a necessary and sufficient condition for the general case of three options has not been known until now, there are some different sufficient conditions that are formulated in the literature. In this paper, we provide a necessary and sufficient condition for the existence of a maximum and the uniqueness of the argument that maximizes the value, i.e. for the evaluability of the data in models of these types. By computer simulation, we present the efficiency of the condition, comparing it to the previously known sufficient conditions.

Paper Structure

This paper contains 22 sections, 10 theorems, 74 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The investigated models
  3. Thurstone motivated models (THMMs)
  4. Two-option models (THMM2)
  5. Three-option Thurstone motivated models (THMM3)
  6. Davidson's model (D3)
  7. Conditions of the existence and uniqueness of the maximizer
  8. Previously known sufficient conditions
  9. Two-option model
  10. Three-option Thurstone motivated models
  11. Davidson's model
  12. The main result: necessary and sufficient condition of data evaluability in THMM3 and D3
  13. Efficiency of the necessary and sufficient condition
  14. Summary
  15. Appendix
  16. ...and 7 more sections

Key Result

Theorem 1

Let $F \in \mathbb{F}$. The maximal values of the log-likelihood functions (eq:LOGLIKL3) and (eq:loglikelihood) exist and the argument is unique under the conditions $m_1=0$, $0<d$ in THMM3 and $0<\underline{\pi },$$0<\nu$, $\sum_{i=1}^{n}\pi_i=1$ in D3, respectively, if and only if the data matrix

Figures (3)

  • Figure 1: The options and the intervals belonging to them in a two-option model.
  • Figure 2: The options and the intervals belonging to them in a three-option model.
  • Figure 3: Rates of fulfillment of the different condition systems related to the necessary and sufficient conditions.

Theorems & Definitions (20)

  • Definition 1: Graph of comparisons belonging to the data in two-option models
  • Definition 2: Graph of comparisons in 3-option models
  • Definition 3: Directed graph belonging to the 'better' options in three-option Davidson's models
  • Definition 4: graph $GR_{dir}^{(3)}$
  • Definition 5
  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Definition 6: Weighted graph $GW$
  • ...and 10 more