what are stakes in cost-benefit terms? Clarifying five examples/contentions

Question

Please excuse me if this is off-topic, and don't be offended if it is stupid.

Scholars theorize that the stakes matter for rational decision making or cost-benefit calculations (under uncertainty or risk).

But I haven't found a reference clearly explaining how the concept of stakes is understood, and how it enters the terms of the calculus, B and C (are stakes considered a B or a C?). I have come across the concept as if it is known and not needed to be interpreted in formal terms (perhaps this is because there is no reason to do so - something I am not aware of or have yet to understand). These are some of the examples I am thinking about but haven't seen discussed or explained clearly in literature but would appreciate clarification of:

• do stakes affect—increase or decrease—B, C, or both e.g. B-C? Are there strong reasons to think about it in one of these ways?
• if one’s anticipated gain is another’s loss: how is it translated into B and C? Both agents have something at stake. But are the terms so encompassing that one’s gain is possible B while another’s loss is possible C?
• An agent might lose what’s at stake irrespective of the initial calculus. So after the fact one might have lost and another might have benefitted. Before the fact, which is the time of decision making or the reason for action, the stakes may be thought of as benefits.
• is it up to the individual agent to judge what the stakes are? Perhaps because one cares more deeply than the other. How are the terms affected? Is caring deeply the same as perceiving high stakes and thus 0<B-C?
• is the interpretation of stakes in formal terms mere semantics or does it have theoretical interpretations i.e. for debates about rational choice, expected utility, prospect theory (loss'/gain's domains), decision making under risk (known outcomes and probabilities of outcomes) vs. uncertainty (unknown probabilities of outcomes), game theory, etc.?

Can you either please clearly explain or refer me to a reference that does that? (or refer me to a better forum or Q&A site for this). My immediate view tends to be that stakes are encompassed as benefits rather than costs. But there may be reasons you can help me understand to not view it like that or to say that an alternative view is equivalent or preferable. Thank you.

Answer 1

For clarity, let's first understand what "stake" means. According to Merriam-Wester, the word "stake" has several meanings, the most relevant to economics is the following:

3 a: something that is staked [i.e. betted or wagered] for gain or loss
b: the prize in a contest
c: an interest or share in an undertaking or enterprise

Whether a stake counts as benefit or cost depends on which sense the word is used in.

a. If stake refers to a wager for potential gain or loss, it can be viewed as a cost.

b. If stake refers to the prize in a contest, then it's a benefit.

c. If stake refers to a share or an interest in something being consequential (financially or otherwise) to a stakeholder, then it is essentially a lottery ticket (in the sense used in expected utility theory), which may be a benefit or a cost depending on its expected value (ex ante) or realization (ex post). In other words, it can be either, depending on the context.

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Now, to the items on which you seek clarifications:

do stakes affect—increase or decrease—B, C, or both e.g. B-C? Are there strong reasons to think about it in one of these ways?

It depends on the context, particularly the sense in which the word "stake" is used. See discussion above.

if one’s anticipated gain is another’s loss: how is it translated into B and C? Both agents have something at stake. But are the terms so encompassing that one’s gain is possible B while another’s loss is possible C?

If $i$'s gain is $j$'s loss, then we simply write $B_i = C_j$, where $B_i$ enters positively in $i$'s utility and $C_j$ enters negatively in $j$'s utility.

An agent might lose what’s at stake irrespective of the initial calculus. So after the fact one might have lost and another might have benefitted. Before the fact, which is the time of decision making or the reason for action, the stakes may be thought of as benefits.

This is consistent with the view of stake as lottery, whose value may differ between ex ante and ex post.

is it up to the individual agent to judge what the stakes are? Perhaps because one cares more deeply than the other. How are the terms affected? Is caring deeply the same as perceiving high stakes and thus 0<B-C?

For stake-as-a-wager or prize, there is usually little room for individual differences in how it is perceived. But for stake-as-a-lottery, individual differences in perception can exist, for example, due to differences in access to information, psychological predispositions, etc. While wagers/prizes are sure outcomes (i.e. there's no uncertainty of what they are), lotteries consist of a set of outcomes and their associated probabilities. It is the probabilities that can be perceived differently by different individuals. Objective probabilities can be misperceived a la Kahneman-Tversky/prospect theory, and subjective probabilities are, well, subjective. That's why wagers/prizes cannot be perceived differently, but lotteries can.

Now, "caring deeply" should be thought of as having an intense preference for or a high valuation of an outcome, and can be modeled with a cardinal utility such as quasilinear or vNM utility. But whether it can be interpreted as "perceiving high stakes" will have to be determined by further details from the context.

Answer 2

HerrK. answered

do stakes affect—increase or decrease—B, C, or both e.g. B-C? Are there strong reasons to think about it in one of these ways?

It depends on the context, particularly the sense in which the word ‘stake’ is used.

Whether a stake counts as benefit or cost depends on which sense the word is used in.

I would like to add that the three meanings of stakes don’t differ much considering that all regard 1) a probability distribution on outcomes (chance/risk of reward) and 2) outcomes (what is there to win/not win/lose).

It is not useful to focus too much on a prize itself (an outcome) because that compares less directly with e.g. a lottery. Conversely, the contest for a prize is comparable with a lottery.

The exception regards wagers/bets i.e. risking to jeapordize current (privately held, personal) value for the possibility of gaining more value (handed by the losing party) or the honor of not losing. In that case, the outcome itself seem to differ from outcomes like a prize or an outcome of a lottery, which are mostly not already private but rather up for grabs.