A Compact, Logical Approach to Large-Market Analysis

Yannai A. Gonczarowski; Scott Duke Kominers; Ran I. Shorrer

Working Paper | HBS Working Paper Series | 2019

A Compact, Logical Approach to Large-Market Analysis

by Yannai A. Gonczarowski, Scott Duke Kominers and Ran I. Shorrer

Abstract

In game theory, we often use infinite models to represent "limit" settings, such as markets with a large number of agents or games with a long time horizon. Yet many game-theoretic models incorporate finiteness assumptions that, while introduced for simplicity, play a real role in the analysis. Here, we show how to extend key results from (finite) models of matching, games on graphs, and trading networks to infinite models by way of Logical Compactness, a core result from Propositional Logic. Using Compactness, we prove the existence of man-optimal stable matchings in infinite economies as well as strategy-proofness of the man-optimal stable matching mechanism. We then use Compactness to eliminate the need for a finite start time in a dynamic matching model. Finally, we use Compactness to prove the existence of both Nash equilibria in infinite games on graphs and Walrasian equilibria in infinite trading networks.

Keywords: Markets; Analysis; Game Theory;

Language: English

Format: Print

28 pages

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Citation:

Gonczarowski, Yannai A., Scott Duke Kominers, and Ran I. Shorrer. "A Compact, Logical Approach to Large-Market Analysis." Harvard Business School Working Paper, No. 19-127, June 2019.

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We introduce a new model of school choice with reserves in which a social planner is constrained by a limited supply of reserve seats and tries to find an optimal matching according to a social welfare function. We construct the optimal distribution of reserves via a quartic-time dynamic programming algorithm. Due to the modular nature of the dynamic program, the mechanism is strategy-proof for reserve-eligible students.

This case is set in mid-2019, when Qualcomm was struggling with unwanted takeover battles, fights with Apple and the Chinese government, and internal dissension on the board of directors. Ten years earlier Qualcomm was hailed as a monopoly on CDMA technologies and its derivatives—it appeared to be positioned to become the next dominant technology company in the tradition of Microsoft and Intel. But as the industry changed, Qualcomm’s competitive position seriously weakened. The purpose of the case is to explore central questions in strategy formulation: 1. how to build a long-term sustainable position based on technology, 2. how to design an IP strategy to lock in customers for the long run, 3. how to build a global standard and appropriate returns on that standard, and 4. how to respond when your historical business model starts to crumble.

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A Compact, Logical Approach to Large-Market Analysis

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