analysis and we have found that dual process cooperation states are the only stochastically stable states in the entire parameter space in which they are absorbing states. This in turn implies that in our model intuitive cooperation is favored by evolution in a larger parameter space than in Bear and David G Rand (2016). This result implies that an endogenous interaction structure promotes cooperation.
Chapter 4
A Dual Process Model for the Evolution of Fair Splits and Harsh Rejections in the Ultimatum Game
4.1 Introduction
Since the first experimental work dealing with the Ultimatum Game (G ¨uth, Schmittberger, and Schwarze, 1982) there has been great interest in un-derstanding why subjects’ behavior differs so markedly from standard game theoretical arguments. In fact, while the latter predicts almost null offers and full acceptance of any positive offer, experimental evidence points to ’fair’ offers – with the average offer being frequently between 40%and 50% of the amount to split – and frequent rejection of ’unfair’
offers (G ¨uth and M. G. Kocher, 2014).
However, the disagreement between theoretical predictions and ex-perimental evidence might not be as strong as it might seem at a first glance. In fact, it is essential to remember that theoretical predictions are derived under many assumptions, and their validity is restricted to the situations in which such assumptions hold. In the case of an Ultimatum
Game, for example, players must believe that the game is anonymous and one-shot. Still, they must also be convinced that the receiver has no bargaining power other than being able to reject the offer. But what if this is not the case?
With this reasoning in mind, I develop a model of the evolution of Ultimatum Game bargaining in which proposers and receivers interact over time in either Ultimatum Games or simplified Bargaining Games.
Hence, agents face different kinds of interactions. Following dual pro-cess theories of cognition (Evans and Stanovich, 2013) and their recent applications to the case of cooperation (Bear and David G Rand, 2016;
Jagau and Veelen, 2017), I assume that although agents know the prob-ability with which they are playing one game or another, they do not know the actual game they are playing in a given interaction. To obtain this information, agents need to pay a deliberation cost which is ran-domly sampled from a known and fixed distribution. Both proposers and receivers can choose their cognitive effort, and they do so by select-ing a threshold cost of deliberation, which represents the maximum cost they are willing to pay to deliberate. In a given interaction, if an agent pays the deliberation cost, then it deliberates, and it gets to know the type of game it is currently facing and, consequently, it can condition its behavior to the game played; while if the agent does not deliberate, then it must play without knowing the actual game. Both proposers and re-ceivers update their strategies according to reinforcement learning. More precisely, each agent is characterized by sets of propensities, with each propensity expressing the agent’s attitude to choose a given action, such as selecting a given threshold cost of deliberation, making a given offer, and accepting or rejecting an offer. After every interaction, each agent revises its propensities according to Roth-Erev reinforcement (Erev and Roth, 1998).
To my knowledge, this is the first model analyzing the evolution of Ultimatum Game bargaining within a dual process cognition framework.
This is of great interest because – in the attempt to understand whether differences between theoretical predictions and experimental findings in the Ultimatum Game are driven by intuitive behaviors – many
experi-mental studies have applied cognitive manipulations to subjects playing the Ultimatum Game (see Capraro (2019) for a review). Some papers find intuition increasing (or equivalently deliberation decreasing) both the of-fers made by proposers (Cappelletti, G ¨uth, and Ploner, 2011; Achtziger, Al ´os-Ferrer, and Wagner, 2016; Halali, Bereby-Meyer, and Ockenfels, 2013) and receivers’ rejection rates (Sutter, M. Kocher, and Strauß, 2003;
Knoch et al., 2006; Cappelletti, G ¨uth, and Ploner, 2011; Grimm and Men-gel, 2011; Neo et al., 2013; Achtziger, Al ´os-Ferrer, and Wagner, 2016).
However, some other works suggest that cognitive manipulations gener-ate either no effects (Clare Anderson, 2010; Cappelletti, G ¨uth, and Ploner, 2011) or even opposite effects to the ones just mentioned (Clare Ander-son, 2010; Halali, Bereby-Meyer, and Ockenfels, 2013). However, despite these unclear results in the experimental literature, no theoretical work has directly tackled the issue.
Some theoretical papers have suggested other possible explanations for the experimental evidence on Ultimatum Game bargaining. For ex-ample, reputation (Martin A Nowak, Page, and Sigmund, 2000; Andr´e and Baumard, 2011), structured interactions (Page, Martin A Nowak, and Sigmund, 2000; Alexander, 2007), and mistakes (David G. Rand, Tar-nita, et al., 2013) have been argued to give rise to fair offers. A more com-prehensive review and classifications of models providing explanations for the evolution of fairness in the Ultimatum Game can be found in De-bove, Baumard, and Andr´e (2016). These models are certainly of great interest. However, they seem to provide only partial explanations of the phenomenon. In fact, these contributions either cannot fully explain why experimental subjects deviate from theoretical predictions in one-shot and anonymous Ultimatum Game bargaining, which do not feature elements such as reputation and structured interactions; or they rely on relatively high mistake probabilities.
On the one side, I find that receivers consistently end up adopting the same strategy: they do not accept an offer unless it is as large as their outside option, which depends on their cognitive contingency (intuition, deliberation if the game is an Ultimatum Game, and deliberation if the game is a simplified Bargaining Game). Moreover, receivers’
delibera-tion patterns are significantly affected by the offer made by the proposer.
In fact, while null offers or generous offers, i.e., offers equal or above 50%of the value to split, always make receivers deliberate relatively in-frequently; medium to low offers push receivers to deliberate more, and this especially holds for given offer-specific probabilities that the game is a simplified Bargaining Game. This implies that proposers’ behavior can significantly impact receivers’ reliance on deliberation (endogenous receivers’ deliberation).
On the other side, I find that proposers follow different strategies de-pending on the width of the cost distribution considered. More precisely, if the cost distribution has large support (and, so, on average high delib-eration costs must be paid to guarantee frequent delibdelib-eration), proposers adopt a pooling strategy according to which they always make the same offer, which is equal to receivers’ intuitive outside option. Instead, if the cost distribution is narrow, proposers switch to the following differenti-ated strategy: if they deliberate and find out that the game is a simpli-fied Bargaining Game, they make an offer equal to the receivers’ outside option under deliberation if the game is a simplified Bargaining Game;
otherwise, they offer to receivers their intuitive outside option. Further, I find that proposers’ deliberation patterns mainly depend on the pro-posers’ own strategy. More precisely, proposers tend to deliberate more frequently the more their strategy is differentiated, which happens for intermediate values of the probability that the game is a simplified Bar-gaining Game.
Finally, the model predicts huge variability in rejection rates condi-tional on a given offer being made by proposers. Such heterogeneity can be found both across offers made and for a given offer as different prob-abilities of playing a simplified Bargaining Game are considered. More-over, even in the absence of mistakes, the model generates rejections rates of about 5% − 10%.
The paper provides a new theoretical explanation for many findings in the experimental literature employing the Ultimatum Game, such as proposers making fair offers, receivers frequently rejecting strictly posi-tive offers, and possibly the existence of cross-country differences in
be-havior. Further, the model provides interesting predictions about the effects of cognitive manipulations in the Ultimatum Game and further suggests new research questions regarding proposers’ and receivers’ be-havior and their strategic interaction in the Ultimatum Game.
The remaining part of the paper is structured as follows: in Section 4.2 I present the theoretical framework considered, in Section 4.3 I illustrate and discuss the main results, and Section 4.4 concludes and provides insights for future research.