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Background

For my thesis, I'm trying to wrap my head around a relatively small field within mathematics called "Stochastic Cooperative Game Theory". To that end, I've read a few papers on the subject, including Cooperative games with stochastic payoffs (Suijs et al., 1999, link) and Convexity in stochastic cooperative situations (Timmer et al., 2005, link).

In classical cooperative game theory, the payoffs (rewards) to players are deterministic. The players know beforehand what the benefits are of forming coalitions with the other players. In stochastic cooperative game theory, however, uncertainty is introduced. This means players do not know for sure what payoff they will receive: the payoffs are modeled as random variables.

The players try to decide which payoffs based on random variables are most beneficial to them. However, there many ways to define "most beneficial". In order to determine which random variable is preferred by player $i \in N$, Timmer et al. introduce the notion of a preference relation $\succeq_{i}$ (on page 4). For some stochastic payoffs $X$ and $Y$ (where $X$ and $Y$ are nonnegative random variables with finite expectation), they write $X \succeq_{i} Y$ if player $i$ weakly prefers the stochastic payoff $X$ to receiving $Y$. Furthermore, $X \succ_{i} Y$ means that player $i$ strictly prefers $X$ to $Y$.

On page 5, the same authors give an example of such a preference relation. The preferences of player $i$ could be such that $X \succeq_{i} Y$ if and only if $E(x) \geq E(Y)$, where $E(X)$ is the expectation of $X$. I suppose I understand this type of preference. As a player, you'd want to maximize the expectation of your payoffs $(*)$.

However, I don't understand the second type of preference. It involves quantiles of random variables. They define $ u_{ \beta_{i} }^{X} = \sup \{t \in \mathbb{R} | \Pr \{ X \leq t \} \leq \beta_{i} \} $ be the $\beta_{i}$-quantile of $X$. Furthermore, they define the utility function $U_{i}$ by $U_{i} (X) = u_{ \beta_{i} }^{X} $. They say that a player has quantile-preferences if $X \succeq_{i} Y$ if and only if $U_{i} (X) \geq U_{i} (Y)$.

Questions

  1. Could you provide me with an intuitive explanation of the quantile preference relation? (Perhaps similar to the one I gave myself in $(*)$, but a bit longer/detailed?). Can you give an example with actual distributions?

Suijs et al. introduce a similar preference relation in their paper on p. 197. They add that agent $i$ is more risk averse than agent $j$ if and only if $\beta_{i} < \beta_{j}$

  1. Could you explain and motivate this definition of risk aversiveness? And how does it relate to this type of risk aversiveness?

Finally, I tried understanding the quantile preference relation by comparing two cumulative distribution functions. These are the cumulative exponential distribution with $\lambda = 0.5$ (distribution $X$) and $\lambda = 1.5$ (distribution $Y$), respectively. Suppose we take $\beta_{i} = 0.3$. As $E[X] = 1/0.5 = 2 > 0$ and $E[Y] = 1/1.5 = 2/3 > 0$, we need to find the utility with $1-\beta_{i} = 0.7$. Then we get the following picture:

This was adapted from an <a href="https://en.wikipedia.org/wiki/Exponential_distribution#/media/File:Exponential_cdf.svg">image</a> on Wikipedia

(adapted from an image on Wikipedia). I suppose now we have $u_{ \beta_{i} }^{X} \approx 2.3$ and $u_{ \beta_{i} }^{Y} \approx 0.9$, so we have $U_{i} (X) \geq U_{i} (Y)$

  1. How do we interpret this result? Is there an intuitive explanation?
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I am by no means an expert in this field but have had to consider this kind of preference before. Consider the following two wagers:

  • Wager A: With probability $p = {{1}\over{1 \times 10^6}}$ you will win $1 \times 10^7$ dollars, and with probability $1-p$, you lose $1$ dollar.

  • Wager B: With probability $p = {{9}\over{10}}$ you will win $10$ dollars and with $1-p$ you will lose $1$ dollar.

It is easy to compute that the expected value of Wager A exceeds that of Wager B. However, for any quantile ${1\over{10}} \lt \beta \lt 1-{{1}\over{1 \times 10^6}}$, Wager B is preferred to Wager A on a quantile basis. On quantiles $\beta \leq {1\over{10}}$, the two wagers are equivalent and on quantiles $\beta \geq 1-{{1}\over{1 \times 10^6}}$, Wager A is preferred.

Essentially, quantile preference is, indeed, a way to model a certain kind of risk-aversion/seeking although I'm not exactly sure how to relate it to the standard definitions as outlined in the wiki article. Intuitively, I would say that if someone is seeking to optimize a high quantile outcome, they're looking to score a long-shot; if someone is seeking to optimize a low quantile outcome, they're looking to avoid a worst-case scenario.

Regarding your example of the two exponential distributions, I'm not sure that it is particularly illuminating since $X$ appears to be preferred to $Y$ in both expectation and at all quantiles $\beta$. Perhaps there might be something to be said regarding the magnitude by which $X$ is preferred to $Y$ at different quantiles versus expected value?

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N.B. I will occasionally call random variables “lotteries” or “gambles”. This is standard in the literature, so I hope you are comfortable with the terminology.

Explaining the Quantile Preference Relation

Quantile preferences are similar to expected-value preferences ($ X \succeq_i Y \iff E(X) \ge E(Y) $) in that they both depend only on a summary statistic of the distribution of a random variable.

The only difference is that while expected-value preferences depend only on the means of the distribution, quantile preferences depend on a specific quantile of the distribution.

Observe that the $\beta_i$-quantile is essentially the inverse of the CDF of the distribution. This means that the $\beta_i$-quantile of a random variable $X$ is the largest value $u_{\beta_i}^X$ such that $\Pr (X \le u_{\beta_i}^X) \le \beta_i$. (For continuous distributions, you can replace the $\le$ with $=$.)

For example, the $\frac 1 2$-quantile is the median of the distribution. If an agent has $\frac 1 2$-quantile preferences, then they only care about what the median realisation of a random variable is. Similarly, the $\frac {1} {10}$-quantile is the first decile of the distribution, and so on.

Risk Aversion

Suijs, et al. say (also on p. 197) that $i$ is more risk averse than $j$ whenever, for all random variables $X$,

$$ \{ Y \vert Y \succeq_i E(X) \} \subset \{ Y \vert Y \succeq_j E(X) \} .$$

In other words, whenever $i$ prefers some lottery $Y$ to some sure amount $E(X)$, $j$ also prefers the lottery $Y$ to $E(X)$. Loosely, you can think of this as $j$ always being willing to gamble whenever $i$ is willing to do so. This has a natural interpretation as $j$ preferring risk more than $i$ does, or that $i$ is more risk-averse than $j$.

Suijs, et al then go on to assert that this definition is satisfied if and only if $\beta_i < \beta_j$. This step isn’t obvious to me, but perhaps this is a result that is well-known by people who work in the field. I’ll try to think more about this and edit my answer if I get somewhere.

The Wikipedia page you link on risk-aversion is related, but has more to do with what Suijs, et al call “absolute risk aversion”. Notice that the Wikipedia page emphasises categorising agents as risk-averse, risk-neutral, or risk-loving (consistently with the definitions that Suijs, et al give for these terms on p. 197). However, as Suijs, et al note, agents with quantile preferences are not risk-averse, risk-neutral, nor risk-loving. However, one agent with quantile preferences can be more or less risk averse than another agent also with quantile preferences.

Your Example

I think there seems to be an issue with your calculation, as you seem to have calculated the utility for an agent with $0.7$-quantile preferences, not $0.3$-quantile preferences. However, it does look like an agent with $0.3$-quantile preferences will prefer lottery $X$ to lottery $Y$ as well.

The intuition of this result is easy. Observe that the CDF of $X$ is always below the CDF of $Y$. (In more formal language, $X$ first order stochastically dominates $Y$.) This means that $Y$ is always (and uniformly) relatively more likely to realise as one of the values on the lower (and hence worse) end of the distribution, and, conversely, $X$ is relatively (uniformly) more likely to realise as one of the values on the higher (better) end of the distribution.

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