Abstract
The goal of the present paper is to construct a formal explication of the pluralistic ignorance explanation of the bystander effect. The social dynamics leading to inaction is presented, decomposed, and modeled using dynamic epistemic logic augmented with ‘transition rules’ able to characterize agent behavior. Three agent types are defined: First Responders who intervene given belief of accident; City Dwellers, capturing ‘apathetic urban residents’ and Hesitators, who observe others when in doubt, basing subsequent decision on social proof. It is shown how groups of the latter may end in a state of pluralistic ignorance leading to inaction. Sequential models for each agent type are specified, and their results compared to empirical studies. It is concluded that only the Hesitator model produces reasonable results.
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Notes
For a comprehensive walk-through of this explanation and supportive data, refer to Latané and Darley (1970).
Halbesleben and Buckley (2004) provides an illuminating overview of the history and development of the term.
Technically, no logic is introduced; the dynamics are investigated using only model theory.
Reflexive and transitive binary relation where every non-empty subset has a minimal element, cf. (Baltag and Smets 2008).
The relation \(\le _{i}\) may therefore more appropriately be thought of as an implausibility relation, where \(s\le _{i}t\) is read ‘\(t\) is as implausible as \(s\), or more so’.
A heuristic to aid recall is that \(s<t\) in form is similar to \(s\leftarrow t\) when looking at the arrowhead. When \(s\le _{i}t\) and \(t\le _{i}s\), arrowheads are omitted altogether.
The notation for information and plausibility cells are adopted from Dégremont (2010).
Parentheses will be omitted where no confusion should arise.
The reason for using this atypical definition of propositions is that it allows us to speak about the same proposition across multiple models. This is practical as model transformations will play a large role in the latter. A Kripke model valuation may easily be extracted from set of doxastic atomic propositions; see (Baltag and Smets 2008) for details.
Ontic facts are all non-doxastic facts, i.e. propositions that do not contain belief or knowledge operators.
The postcondition \(\top \) leaves all atomic propositions as they were in the previous model. This is specified by the action priority update product below.
\(\preceq _{i}\) is from \(\mathbf {E}\) and \(\le _{i}\) from \(\mathbf {S}\). \(\sigma \prec _{i}\tau \) denotes (\(\sigma \preceq _{i}\tau \) and not \(\sigma \succeq _{i}\tau \)), \(\sigma \backsimeq _{i}\tau \) denotes (\(\sigma \preceq _{i}\tau \) and \(\sigma \succeq _{i}\tau \)).
Note the analogy with numerical equations; for both \(2+x=5\) and \(\{2+x=5,4+x=7\},\,x=3\) is the (unique) solution.
The definition is altered to suit transition rules using \([X]_{i}\) “modalities” by suitable replacing \([X]\) with \([X]_{i}\) and \([\varGamma ]\) with \([\varGamma ]_{i}\) throughout.
For a definition of system, see (Rendsvig 2013a).
Unless evading entails leaving the scene or observation is performed in a non-discrete manner. Often this is not the case, though: “Among American males it is considered desirable to appear poised and collected in times of stress. ... If each member of a group is, at the same time, trying to appear calm and also looking around at the other members to gauge their reactions, all members may be led (or misled) by each other to define the situation as less critical than they would if alone. Until someone [intervenes], each person only sees other nonresponding bystanders, and ... is likely to be influenced not to act himself.” (Latané and Darley 1968, p. 216); “... Apparent passitivity and lack of concern on the part of other bystanders may indicate that they feel the emergency is not serious, but it may simply mean that they have not yet had time to work out their own own interpretation or even that they are assuming a bland exterior to hide their inner uncertainty and concern.” (Latané and Rodin 1969, p. 199).
The second quote in the previous note seems to indicate the plausibility of this assumption.
When constructing APU products, a state in the product model is an ordered pair \((s,\sigma )\) of a state \(s\) and and action \(\sigma \). In this pair, \(s\) may again be such a pair. Say that a predecessor of \(s'\) is any \(s\) that occurs in any of the ordered pairs of \(s'\), including \(s'\) itself.
It is possible to give agents a choice of interpretation by invoking transition rules with interpretation rules as possible solutions. In the present, agents are given no choice of interpretation, and this construction is consequently skipped for simplicity.
Each \({\mathbf {E}_{\mathbf{2}}}_{j}\) functions as a truthful public announcement of \(pre(\rho )\), for which the order of announcements does not matter (Baltag and Smets 2009): states are deleted, the remaining orderings staying as previous. Deleting simultaneously or in some sequence makes no difference.
Though time has passed, beliefs have not changed, and this is known to all: \(\mathbf {S}_\mathbf{3}\models \bigwedge _{i\in \mathcal {A}}K_{i}(\bigwedge _{j\in \mathcal {A}}[\mathbf {S}_{\mathbf{1}}]B_{j}\varphi \rightarrow B_{j}\varphi )\) for \(\varphi \in \{A,\lnot A\}\).
Strictly speaking, in the present model agents do not revise their beliefs. An additional operator is instead introduced to facilitate comparison with private beliefs. A belief revision policy may easily be defined using decision rules to the effect that agents update their beliefs under the suitable circumstances, see (Rendsvig 2013b).
Concerning \(\mathbf {E}_{\mathbf{0}}\), it should be obvious how the APM must be altered to include further agents, while maintaining complete higher-order ignorance.
Again, it should be obvious how \(\mathbb {E}_{\varGamma }\) may be altered to accommodate for a larger population.
The shift was made to ease the exposition. Influenced agents require the notion of social beliefs, not necessary for Hesitators’ first choice.
The requirement that \(i\) must have observed before acting on social beliefs ensures that agents do not intervene immediately after seeing the accident (a private belief that \(A\) would imply that \(SB_{i|G}A\), as agents then hold no beliefs regarding others’ beliefs).
Cf. the tie-breaking rule used in the definition of social beliefs.
See Latané and Darley (1968) for calculation of hypothetical baseline based on the alone condition.
In a mixed population model, using City Dweller agents for the two confederates.
Again in a mixed population model, using City Dweller agents for the two confederates.
How well these individual studies conform to the pluralistic ignorance explanation of the bystander effect has not been checked.
I.e., interchanging the \(\alpha ,\beta \) tie-breaking parameters. Alternatively, social beliefs could be defined by weighing others’ perceived beliefs higher than one’s own, or by moving to a threshold rule requiring e.g. perceived agreement with all peers as done in Seligman et al. (2013), Christoff and Hansen (2013).
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Acknowledgments
The author would like to thank the editors for organizing the CPH-LU workshops on social epistemology, as well as the participants of said workshops for valuable comments and discussion. The work has benefited especially from discussions with Henrik Boensvang and Vincent F. Hendricks. Carlo Proietti is thanked for his corrections to the manuscript. Finally, a warm Thank You to the two anonymous reviewers: the comments, criticisms and correction provided by your thorough reading of the original submission have been invaluable.
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Rendsvig, R.K. Pluralistic ignorance in the bystander effect: informational dynamics of unresponsive witnesses in situations calling for intervention. Synthese 191, 2471–2498 (2014). https://doi.org/10.1007/s11229-014-0435-0
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DOI: https://doi.org/10.1007/s11229-014-0435-0