Abstract
In game theory, recently models and solution concepts of games with unawareness have been developed. Focusing on static games with unawareness, this paper discusses generalized Nash equilibrium, an existing equilibrium concept. Some generalized Nash equilibria can be unstable in the sense that, once an equilibrium is played, some agent’s belief is falsified at some level of someone’s perception hierarchy. Based on the observation, we characterize a particular class of generalized Nash equilibrium that expresses stable belief hierarchies so that it can avoid such a problem. This class of equilibrium can be motivated as a stable convention of the game. We also study how unawareness can affect the agents’ behaviors in a stationary state.
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Notes
In this regard, Schipper (2014) pointed out, “equilibrium notions in strategic situations with unawareness may make sense only in special situations such as when players’ awareness along the equilibrium path never changes, or when becoming aware also implies magically also mutual knowledge of the new equilibrium convention.” Halpern and Rêgo (2014) also wrote, “While we think that an understanding of these generalized Nash equilibrium will be critical to understanding solution concepts in games with awareness ... we are not convinced that (generalized) Nash equilibrium is necessarily the “right” solution concept ... this still leaves open the question of what is the “right” solution concept.”
Therefore, we admit that, if we simply understand conventions as states from which no one would like to deviate, our requirement may be too strong to capture the notion exactly. (For example, if the agents have dominant actions, they would always use them, and we do not need to care about consistencies of their beliefs.) But the definition of convention is controversial. Traditionally Lewis (1969) requires common knowledge of conjectures (which is essentially the same as stable belief hierarchies), while some recent arguments say that it is too restrictive (e.g. Binmore 2008). We view that our notion of stable belief hierarchies would work at least as a sufficient condition for stable conventions: If, in a repetitive situation, the agents play a generalized Nash equilibrium and their conjectures are common knowledge, then it can naturally be considered as a stable convention.
The idea behind the concept is very close to that of awareness equilibrium in an unpublished paper by Copic and Galeotti (2007). In their framework, perception hierarchies are not given exogenously, and the definition of equilibrium requires the agents to have perceptions that are consistent with the realized outcome. On the other hand, most studies of games with unawareness, including the present study, formulate awareness structures given exogenously and define solution concepts.
In one-shot games, wrong beliefs can often happen. In standard games, this motivates the notion of rationalizablity, a weaker concept than Nash equilibrium. Heifetz et al. (2013) extend it to games with unawareness.
In terms of games with unawareness, a standard game with common knowledge is a special case, where Nash equilibria coincide with generalized Nash equilibria with stable belief hierarchies.
Kaneko (2002) discusses such a false belief in terms of epistemic logic.
C3 is equivalent to say that \(f_i(G^k) = f_i \circ f_i (G^k)\) for any \(G^k \in \mathcal {G}\) and \(i \in N\). That is, at \(G^k\), if agent i views some game, then she believes she views the game. This is close to positive introspection in epistemic logic.
In game theory, a strategy is often interpreted not as an agent’s choice but as someone’s belief (about the agent’s choice). We follow this interpretation.
This is because, even when \(\delta _i \equiv \delta '_i\), \(\delta _i\) and \(\delta '_i\) may be probability distributions to different sets.
Moreover, in some cases, she may be forced to revise her perception hierarchy as well. For example, in \(\sigma ^*_2\), Alice uses \(a_3\), which was not common knowledge before the play. Thus it is natural to consider that, after the play, the availability of the action becomes common knowledge. That is, the agents would update their perception hierarchies so that \(a_3\) is common knowledge. This means the “graph” (such as Fig. 2) per se can change. While, in general, it is far from clear when an agent updates her perception hierarchy (i.e. the condition of perception revision), we consider the instability of belief hierarchies is a necessary condition of updating it. This is based on an intuition that, if everything is just as expected, then the agent would have no reason to change her perception hierarchy. Therefore we focus only on stability of belief hierarchies in this paper.
The mechanism of belief revision is beyond the scope of this paper. The issue may be related to dynamic epistemic logic.
Generalized Nash equilibrium in static games with unawareness is equivalent to a special case of Bayesian equilibrium in Bayesian games with unawareness of Meier and Schipper (2014). In a Bayesian game with unawareness, states are defined in terms of an unawareness belief structure of Heifetz et al. (2006) and, for every agent, a type mapping associates each state with probabilistic belief over some states. Then, in a Bayesian equilibrium, for every type of every agent, a best response is taken in light of its belief. Thus, according to Remark 1, adding the requirement of stable belief hierarchies to generalized Nash equilibrium is essentially equivalent to considering a Bayesian equilibrium in which, for every agent, every type’s choice is the same. Meier and Schipper (2014) also use Feinberg’s (2012) example to illustrate their model, so the reader may want to compare our formulation in 2.2 and theirs.
Sasaki et al. (2015) discuss such situations within a game theoretical framework, though not in a very formal way.
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The preliminary version of this paper was presented in The 4th International Symposium on Integrated Uncertainty in Knowledge Modelling and Decision Making (Sasaki 2015). The author thanks Guest Editor Van-Nam Huynh and two anonymous referees for their careful reviewing and helpful comments. This research is supported by KAKENHI Grant-in-Aid for Young Scientists (B) 15K16292.
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Sasaki, Y. Generalized Nash equilibrium with stable belief hierarchies in static games with unawareness. Ann Oper Res 256, 271–284 (2017). https://doi.org/10.1007/s10479-016-2266-5
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DOI: https://doi.org/10.1007/s10479-016-2266-5