# Schedule - Parallel Session 5 - Ambiguity in Games: Theory

WMG IMC Auditorium 0.02 - 14:00 - 15:30#### Ambiguity in Games?

Ken Binmore

#### Abstract

Various modifications of rational choice theory have sought to accommodate reported deviations from Bayesian decision theory in experiments on the Ellsberg paradox. The new theories usually postulate some ambiguity in the probabilities assigned to uncertain events. How well do such theories work when applied in game theory? This question is explored from the viewpoint of Leonard Savage, who argued that his newly created theory of subjective expected utility is only realistically applicable in what he called a small world. The world of classical game theory is small almost by definition, but we endorse the view that the classical notion of a mixed strategy needs to be expanded to allow for devices from the theory of algorithmic randomization that permit pure strategies to be muddled together in a manner that defies a Bayesian description. However, we depart from the developing orthodoxy on game theory in the presence of Knightian uncertainty in also denying that a description in terms of multiple priors is adequate. In particular, we offer an argument against the common use of the maximin criterion in evaluating the payoffs that result when the players use Knightian strategies modeled in terms of multiple priors. The paper continues by offering an argument that favors replacing the standard additive Hurwicz criterion by a multiplicative Hurwicz alternative. An example demonstrates that such a replacement sometimes generates muddled Nash equilibria that are Pareto-improvements on standard mixed Nash equilibria.

**Ken Binmore**

#### Strategic Ambiguity in Two-Player Extensive-Form Games with Perfect Information

David Kelsey

#### Abstract

This paper proposes a solution concept for finite games of complete and perfect information with ambiguity. We model ambiguity with neo-additive preferences. As the game proceeds players gain new information and update their preferences. This updating is modelled by the Generalised Bayesian Updating rule (GBU), Eichberger Grant and Kelsey-(2007). Individuals who deviate from expected utility theory will not necessarily be dynamically consistent. In an extensive-form game this implies that the equivalence of a strategy in the normal form of the game with a sequence of subgame-perfect optimal actions in the extensive form game no longer holds. Just as a player may no longer want to follow the moves of her ex-ante planned strategy relative to her opponents’ behaviour, she may not be able to commit to her own planned actions because her updated beliefs are not additvely separable. Following Strotz (1955), Bose-Renou (2014) and Siniscalchi (-(2011), we give up commitment to a strategy in a sequential decision problem in favour of consistent planning. This implies that at each node where the player has to move, a “sophisticated” player will only consider strategies which remain optimal given her updated future beliefs. Consistent planning means that a player takes into account preference changes due to updating at future nodes. Consistent planning is a sequential decision rule leading to a backward-induction consistent sequence of moves, but not necessarily to an ex-ante optimal strategy with commitment. With consistent planning, however, dynamic consistency is no longer an issue. This may be explained intuitively that players anticipate the new knowledge which they may obtain at future decision nodes and any change in ambiguity this may cause. They then choose the current course of action which maximises anticipated future utility. Alternatively this may be described as players acting strategically against their future selves in an agent-normal form of the game. We illustrate the solution concept by applying it to the centipede game. We find that with ambiguity-aversion the only equilibrium involves playing stop at every node. This is similar to the Nash equilibrium. In contrast with ambiguity-loving it is possible to get an equilibrium in which cooperation continues until nearly the final node. For other parameter values it is possible that cooperation will start but will break down at a random point during the game. This is consistent with the experimental evidence on the centipede game.

**David Kelsey**

#### A Note on Comparative Ambiguity Aversion and Justifiability

Simone Cerreia-Vioglio; Pierpaolo Battigalli; Fabio Maccheroni; Massimo Marinacci

#### Abstract

In this paper we consider a decision maker (DM) who ranks alternatives under uncertainty. The DM holds subjective beliefs over a set of probabilistic models Σ⊆Δ(S), where S is a set of states of nature, or actions of an opponent in a game. We assume that the DM ranks choices according to the smooth ambiguity criterion of Klibanoff et al. (2005). With this, we show that higher ambiguity aversion expands the set of actions that are best replies to at least one belief; for brevity, we call such actions “justifiable.” Empirically, they are the actions that an outside observer can infer as possible from the knowledge of the DM attitudes toward uncertainty. Our result shows that such inference becomes rougher as ambiguity aversion increases. In turn, this implies that higher ambiguity aversion expands the set of rationalizable actions of a game, where the rationalizability concept is modified to take into account ambiguity attitudes. We derive our result from a generalization of the duality lemma of Wald (1949) and Pearce (1984) that should be of independent interest. Another consequence of the same duality lemma is that, under ambiguity neutrality, higher risk aversion expands the set of justifiable actions, and hence the set of rationalizable actions in a game. This risk version of our result was independently obtained by Weinstein (2013) for subjective expected utility maximizers in finite games. For expositional purposes and to exploit economies of scope, we present the results about comparative risk aversion and comparative ambiguity aversion jointly. The result is not intuitively obvious. Indeed, if the DM deems possible very different probabilistic models, then higher ambiguity aversion increases the attractiveness of “safe” actions whose objective expected utility is somewhat low for each model, but does not change much with the model. Given the same belief over probabilistic models, actions that give high expected utility for some models and low expected utility for other models become instead less attractive. Yet, an increase in ambiguity aversion cannot make such actions unjustifiable, because —regardless of ambiguity attitudes— they can always be justified by extreme beliefs assigning high probability to models under which they yield high objective expected utility. This comparative statics result is analogous to another result of ours, which also relies on the smooth ambiguity criterion: higher ambiguity aversion expands the set of self-confirming equilibria (Battigalli et al., 2015). However, the similarity between these results is only superficial, because they rely on different assumptions about the decision or game problem and have very different explanations.

**Simone Cerreia-Vioglio**

#### Social Ambiguity

Uyanga Turmunkh; Richard Gonzalez; Chen Li; Peter Wakker

#### Abstract

Ellsberg (1961) showed that people treat ambiguity in a fundamentally different way than risk. A big field of application for decision making under ambiguity is game theory, where the uncertainty concerns the action of an opponent who interacts with the decision maker, and may have common or opposite interests. Game theory has traditionally used probabilities to model such uncertainties, primarily through mixed strategies. However, in applications the probabilities of opponents’ actions are virtually never available, and game theory typically concerns situations of strategic ambiguity. But also if there is no strategic interaction, ambiguity about moves of an optimizing decision maker with a free will may be perceived differently than ambiguity about moves of nature, which has no interests. In strategic ambiguity there is always a social component. We use the term social ambiguity to distinguish this component. Differences between social situations and nature may be due not to the strategic nature of the uncertainty, but to the mere presence of social ambiguity. To assess strategic ambiguity attitudes, it is warranted to control for social ambiguity attitude. This paper measures subjects’ attitudes to two different sources of social ambiguity and compares these to their attitude toward ambiguity generated by an Ellsberg-urn-type mechanism. We find that subjects are averse to Ellsberg-urn-type mechanical ambiguity, preferring risk to ambiguity of this type. Attitudes toward social ambiguity, however, are reversed: subjects prefer social ambiguity to risk. Thus, we find ambiguity aversion toward mechanical ambiguity but ambiguity seeking toward social ambiguity. Moreover, the observed difference in ambiguity attitudes is wholly attributed to the difference in the motivational (preference) component. When we compare the cognitive (sensitivity) component, we find no difference. This underscores that people have a preference for social ambiguity.

**Uyanga Turmunkh**