De Finetti’s legacy in dealing with uncertainty: Towards finance, artificial intelligence and beyond

Abstract De Finetti’s ideas played a leading role in decision theory and they are still relevant in finance and artificial intelligence. By limiting to unconditional assessments, we focus on the key concept of coherence, that can be formulated either as consistency with respect to an uncertainty model or as fairness in a combination of bets. De Finetti’s approach to uncertainty, originally proposed to justify finitely additive probability theory, naturally leads us to more modern theories of uncertainty, involving non-additive measures and non-linear expectations. We highlight that the link with such theories of uncertainty is twofold: firstly, under suitable logical and numerical conditions, probabilistic coherence gives rise to particular classes of non-additive uncertainty measures and non-linear expectations, in terms of envelopes of coherent extensions and induced previsions; secondly, generalizing de Finetti’s protocols of consistency and betting, coherence can be directly reformulated in such non-additive theories of uncertainty. We show this second approach by developing a coherent theory of $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> -DS Choquet expectations, that are distinguished non-linear expectations, encoding the agent’s attitude towards uncertainty in a fixed pessimism index $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> .