Multivariate Rényi divergences characterize betting games with multiple lotteries

We provide an operational interpretation of the multivariate Rényi divergence in terms of economic-theoretic tasks based on betting, risk aversion, and multiple lotteries. Specifically, we show that the multivariate Rényi divergence D α ̲ ( P ⃗ X ) of probability distributions P ⃗ X = ( p X ( 0 ) , ... , p X ( d ) ) and real-valued orders α ̲ = ( α 0 , ... , α d ) quantifies the economic-theoretic value that a rational agent assigns to d lotteries with odds o X ( k ) ∝ ( p X ( k ) ) − 1 ( k = 1 , ... , d ) on a random event described by p X ( 0 ) . In particular, when the odds are fair and the rational agent is allowed to maximize over all betting strategies, the economic-theoretic value (precisely, the isoelastic certainty equivalent) that the agent assigns to the lotteries is exactly given by w R ̲ ICE = exp [ D α ̲ ( P ⃗ X ) ] , where R ̲ = ( R 1 , ... , R d ) is a risk-aversion vector with R k = 1 + α k /