Dynamic Unstructured Bargaining with Private Information: Theory, Experiment, and Outcome Prediction via Machine Learning

Abstract

We study dynamic unstructured bargaining with deadlines and one-sided private information about the amount available to share (the "pie size"). Using mechanism design theory, we show that given the players' incentives, the equilibrium incidence of bargaining failures ("strikes") should increase with the pie size, and we derive a condition under which strikes are efficient. In our setting, no equilibrium satisfies both equality and efficiency in all pie sizes. We derive two equilibria that resolve the trade-off between equality and efficiency by favoring either equality or efficiency. Using a novel experimental paradigm, we confirm that strike incidence is decreasing in the pie size. Subjects reach equal splits in small pie games (in which strikes are efficient), while most payoffs are close to either the efficient or the equal equilibrium prediction, when the pie is large. We employ a machine learning approach to show that bargaining process features recorded early in the game improve out-of-sample prediction of disagreements at the deadline. The process feature predictions are as accurate as predictions from pie sizes only, and adding process and pie data together improves predictions even more.

Bargaining under time pressure from deadlines

“…As a consequence of the instantaneous nature of the equilibrium prediction, or the intricacies of full-fledged theoretical models taking timing into account, economists have mostly neglected issues of time pressure and deadlines in empirical assessments of bargaining. 2 This paper provides empirical insights based on an experiment in a rich bargaining context (Gächter and Riedl 2005;Karagözoğlu and Riedl 2015;Bolton and Karagözoğlu 2016;Camerer et al 2017) that yet has enough structure to rigorously control for important aspects in bargaining. It extends the scarce existing evidence on the effects of deadlines from simple and highly structured bargaining games such as the ultimatum game (Sutter et al 2003;Cappelletti et al 2011) to a more realistic environment that allows for taking strategic timing decisions of offers and other bargaining parameters explicitly into account.…”

“…Our experiment uses an unstructured bargaining protocol, which allows the sequence and the timing of offers to be endogenously determined. As summarized by Camerer et al (2017), (1) the unstructured bargaining protocol offers the researcher much richer data -especially on the negotiation process -than the structured alternatives, (2) it is more realistic than those, and (3) structured theoretical predictions can still be obtained. Our main treatment variable in this setup is the time allotted to bargainers for reaching an agreement.…”

Real-time tacit bargaining, payoff focality, and coordination complexity: Experimental evidence

“…Both are efficient, but only the latter maximizes total payoffs.5 In his book, The Strategy ofConflict (1960), Thomas Schelling gives a general discussion of tacit bargaining situations (see for example p. 102-108) and describes inSchelling (1961) an experimental design where pairs of subjects tacitly decide which parts of the United States to occupy. Some preliminary experimentation was done but no data were published(Schelling, personal communication).6 InCamerer et al (2015) andGaleotti et al (2016) players make proposals in real time, but earnings are not cumulated over time, and an agreement is assumed to be binding and terminates interaction.…”

Predicting the Replicability of Social Science Lab Experiments

“…For the continuous measure, we study the ratio of effect sizes, standardized to correlation coefficients. Our method uses machine learning to predict outcomes and identify the characteristics of study-replication pairs that can best explain the observed replication results [30][31][32][33].…”