We provide a theory of strategic interactions in static games in the presence of ignorance, i.e., when players cannot produce beliefs as probability measures (or similia) concerning the uncertain elements present in the interaction they are involved in. Assuming only players ' ordinal preferences as transparent, we investigate players that are either optimistic or pessimistic, that we deem tropicalplayers. To explicitly formalize these attitudes , we employ tools from interactive epistemology, by defining the corresponding epistemic events in epistemic possibility structures, which are the counterpart of epistemic type structures suited for our analysis in the presence of ignorance, We show that the behavioral implications related to common belief in these events have algorithmic counterparts in terms of iterative deletion procedures. While optimism is related to Point Rationalizability, to capture pessimism we introduce a new algorithm , deemed Wald Rationalizability . We show that the algorithmic procedure capturing optimism selects a subset of the strategies selected by the algorithmic procedure capturing pessimism. Additionally, we compare both algorithmic procedures to an analogous algorithm based on URLers dominance, deemed URLers Rationalizability, and we show that in generic static games both Point Rationalizability and Wald Rationalizability select a subset of the actions selected by URLers Rationalizability. More generally, while we prove that dropping the genericity assumption does not change the relation between Point Rationalizability and URLers Rationalizability, we show that Wald and URLers Rationalizability are not comparable in their behavioral implications, and we shed light on why this difference emerges. Finally, we explore connections to strategic wishful thinking .
Assuming only ordinal preferences as transparent, we study players interacting under the veil of ignorance, that cannot produce beliefs as probability measures but rather have coarse beliefs represented as subsets of opponents' actions. These players follow either \max \max or \max \min decision criteria. The criteria can be identified as optimistic and pessimistic attitudes, respectively, which we refer to as ``tropical''. Explicitly formalizing these attitudes and how players reason under ignorance, we characterize the behavioral implications related to common belief in these events : while optimism is related to Point Rationalizability, a new algorithm -- Wald Rationalizability -- captures pessimism. Our characterizations allow us to uncover novel connections and results: (i) regarding optimism, we prove that dropping the (implicit) assumption that whatever a player believes is also true allows to capture wishful thinking \`a la Yildiz (2007), thus reversing an existence failure described therein; (ii) by studying how pessimism and optimism relate to URLers dominance, we shed light on the appropriate notion of rationality in ordinal games; (iii) finally, with respect to pessimism, our analysis identifies Wald Rationalizability as the limit point of players becoming infinitely risk averse, hence, clarifying the conceptual underpinnings behind a discontinuity in the analysis of Rationalizability in presence of varying risk attitudes hinted in Weinstein (2016) .
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We provide a theory of strategic interactions in static games in the presence
Assuming only ordinal preferences as transparent, we study players interacting under the veil
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(or similia) concerning the uncertain elements present in the interaction they are involved in. Assuming only players ' ordinal preferences as transparent, we investigate players that are either optimistic or pessimistic, that we deem tropicalplayers. To explicitly formalize these attitudes , we employ tools from interactive epistemology, by defining the corresponding epistemic events in epistemic possibility structures, which are the counterpart of epistemic type structures suited for our analysis in the presence of ignorance, We show that the
but rather have coarse beliefs represented as subsets of opponents' actions. These players follow either \max \max or \max \min decision criteria. The criteria can be identified as optimistic and pessimistic attitudes, respectively, which we refer to as ``tropical''. Explicitly formalizing these attitudes and how players reason under ignorance, we characterize the
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to capture pessimism we introduce
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, deemed Wald Rationalizability . We show that the algorithmic procedure capturing optimism selects a subset of the strategies selected by the algorithmic procedure capturing pessimism. Additionally, we compare both algorithmic procedures to an analogous algorithm based on URLers dominance, deemed URLers Rationalizability, and we show that in generic static games both Point Rationalizability and Wald Rationalizability select a subset of the actions selected by URLers Rationalizability. More generally, while
-- Wald Rationalizability -- captures pessimism. Our characterizations allow us to uncover novel connections and results: (i) regarding optimism,
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genericity assumption does not change the relation between Point Rationalizability and URLers Rationalizability, we show that Wald and URLers Rationalizability are not comparable in their behavioral implications, and
(implicit) assumption that whatever a player believes is also true allows to capture wishful thinking \`a la Yildiz (2007), thus reversing an existence failure described therein; (ii) by studying how pessimism and optimism relate to URLers dominance,
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why this difference emerges. Finally, we explore connections to strategic wishful thinking
the appropriate notion of rationality in ordinal games; (iii) finally, with respect to pessimism, our analysis identifies Wald Rationalizability as the limit point of players becoming infinitely risk averse, hence, clarifying the conceptual underpinnings behind a discontinuity in the analysis of Rationalizability in presence of varying risk attitudes hinted in Weinstein (2016)