The security level models of Gilboa (1988) and of Jaffray (1988) as well as the security and potential level model of Cohen (1992) accomodate succesfully classical Allais paradoxa while they offer an interesting explanation for their occurrence. However, experimental data suggest a systematic violation of these models when lotteries with low probabilities of bad or good outcomes are involved. The present paper develops an axiomatic model that allows for thresholds in the perception of security and potential levels. The derived representation of preferences accommodates the observed violations of the original security and potential level models and provides a natural explanation for their occurence. Additionally, a more fundamental problem of the original models is resolved.
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