**Speaker**: Nina Anchugina (PhD Candidate, University of Auckland)

**Title of the talk**: “A Puzzle of Mixing Discount Functions” joint work with Matthew Ryan (AUT) and Arkadii Slinko (University of Auckland)

**Date, Time and Venue**: Wednesday, 15 March 2017, 2:00-3:00 pm, 206-202 [Arts 1 Building, Level 2]

**What this talk is going to be about, in Nina’s own words**: “This talk will introduce the concept of a “discount function” from decision theory, and discuss some results on mixtures of discount functions. These results suggest a puzzle that we are struggling to resolve. Your help is sought!

In decision-theory, intertemporal preferences are modelled using “discount functions”, which attach weights to different points in time at which costs or benefits might be experienced. In reliability theory, “survival functions” describe the probability that a component survives beyond any given point in time. Discount functions and survival functions have similar mathematical properties. If S(t) is a survival function, its “failure rate” is given by -S'(t)/S(t). For discount functions, the analogous quantity is known as the “time preference rate”. For exponential functions, this rate is constant. For hyperbolic functions, which have become popular for modelling intertemporal preferences, this rate is strictly decreasing – a phenomenon known as strictly decreasing impatience (DI). It is well known that mixing – that is, forming convex combinations – of exponentials produces a function that exhibits strictly DI. (The analogous result is also well known in reliability theory.) We study generalisations of this phenomenon, from which a puzzle emerges. For example, we have not been able to prove (or disprove) that mixing an exponential function with a non-exponential function that exhibits DI will always produce a mixture that exhibits strictly DI. Are we missing something, or do these mixtures behave very strangely?”

Everyone welcome!