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In their seminal works, Arrow (1965) and Pratt (1964) defined two aspects of risk aversion: absolute risk aversion and relative risk aversion. Based on their definitions, we define two aspects of risk: absolute risk and relative risk. We consider situations in which, by making an investment, an agent exchanges a certain amount of wealth w by a random distributed level of wealth w. In such situations, we define absolute risk as the riskiness of a gamble that is distributed as w −w, and relative risk as the riskiness of a security that is distributed as w/w. We measure absolute risk by the Aumann and Serrano (2008) index of riskiness and relative risk by an equivalent index that we develop in this paper. The two concepts of risk do not necessarily agree on which one of two investments is riskier, and hence they capture two different aspects of risk.

It is said that risky asset h acceptance dominates risky asset k if any decision maker who rejects the investment in h also rejects the investment in k. While in general acceptance dominance is a partial order, we show that it becomes a complete order if only infinitely short investment time horizons are considered. Two indices that induce different variants of this order are proposed, absolute acceptance dominance and relative acceptance dominance, and their properties are studied. We then show that many indices of riskiness that are compatible with the acceptance dominance order coincide with our indices in the continuous-time setup.

The hawk-dove game admits two types of equilibria: an asymmetric pure equilibrium in which players in one population play "hawk" and players in the other population play "dove," and a symmetric mixed equilibrium. The existing literature on dynamic evolutionary models shows that populations will converge to playing one of the asymmetric pure equilibria from any initial state. By contrast, we show that plausible sampling dynamics, in which agents occasionally revise their actions by observing either opponents' behavior or payoffs in a few past interactions, can induce the opposite result: global convergence to a symmetric mixed equilibrium.

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We axiomatically characterise two new orders of desirability of gambles (risky assets) that are natural extensions of the proportional stochastic dominance order to complete orders. These orders are represented by indices with parallels to the recently introduced Aumann-Serrano index of riskiness and the Foster-Hart measure of riskiness. The new indices are shown to be related to the concept of coherent measures of risk and to the Sharpe ratio.

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