This came out of session with @vladzamfir earlier this week.
In financial theory, we can roughly approximate how compelling an investment is by comparing the returns (excess of riskfree rate) to a proxy of risk (commonly the standard deviation of the returns i.e. Sharpe ratio).
Therefore, comparing returns / risk is a common way to compare various assets in portfolio theory. However, that approach is often limited to one perspective of what risk is. Therefore, when discussing a heterogenous validator set, Sharpe ratio is far too simplistic to model a validatorâs risk assessment. While we will continue to improve this definition, here is a proposed working model of a validatorâs perception of risk:
\delta_i = \frac{\sigma_{perfect} + \sigma_{error}}{1p_{byzantine}} * (1+b_i)
Perceived risk proxy with respect to (1) risk of the perfect game, (2) unknown risk, (3) perception of byzantine peers, and (4) portfolio concentration risk
where:

\delta_i is a behavioral model of a validatorâs own view of the perceived risk of participation at any given point.
 While a validatorâs perspective may change at any point, it can act only decide to participate, stay or exit. There will be another section that handles withdrawal delay and related costs to staying & exiting (and consequently participating)

\sigma_{perfect} is the theoretical standard deviation of being a validator (i.e. perfect execution).
 Should be same a priori and a posteriori.
 Just using this would result in a Sharpe ratio.

\sigma_{error} is the additional risk due to perceived errors outside of the game (i.e. client bugs, new systems).
 Highest a priori and should asymptotically approach zero a posteriori (validator bugs or commonplace aversion to new processes).

p_{byzantine} is the validatorâs perceived proportion of byzantine validators in the validator set.
 It is a proxy for the common prior assumption in Bayesian games (with incomplete information).
 This will diverge to either a honest supermajority or a byzantine quorum over an iterated game, but the perception of this state on any given round will affect the marginal validatorâs perceived risk.
 \frac{1}{1  p_{byzantine}} can range from 1 when there is full belief that they are honest to larger multiples of risk when people believe there are significant byzantine proportion of actors).
 This magnifies the overall risk. We can tune the relationship with a constant k_0 as well.
 Also, we can replace 1  p_{byzantine} with k_{byz}  p_{byzantine} where k_{byz} is the byzantine quorum threshold of \frac{1}{3}.

b_i is a proxy for portfolio concentration and need for diversification & liquidity. We can begin this by approximating
amount validated / total investment budget
for a given validator (without having optimized, letâs start the framework at [up to double the risk for going âallinâ]). This will model how an investment with the same Sharpe ratio equivalent will make the investment far more risky for someone with a lower total investment budget and therefore makes a given absolute amount investment more risky as a percentage of their portfolio.
 For example, the same $25k angel investment in a startup is exceedingly more risky for someone with $100k vs $100m in wealth. So for a given sharpe ratio, validators will need to be more risktaking to invest in an asset with a higher % of its own investment budget. This proxy reflects that.
 This proxy will come in handy when we discuss heterogenous wealth/income distribution of validators.
While WIP, we can imagine replacing the Sharpe Ratio with this proxy ratio (name tbd, PRR
ratio below; for âPerceived Risk/Rewardâ ratio) that captures various factors that model validator perceived risk.
PRR = \frac{r_v  r_f}{\delta_i}
where high V values represent a more compelling mechanism for validators. (where r_v is risk of validation and \delta_i is defined above. r_f is mentioned for completeness)