Risk aversion, measured in reach units

§ 1The raw order, and the convention it serves

Two players, two rounds, a convex cost $c$ of effort common to both, with $c(0)=0.$ The paper’s results are comparative statics: the lowest live rung $\underline m,$ the highest, the lead intervals, the thresholds — each is a function of the cost, and the question is how it moves when the cost “becomes more convex.” That phrase needs an order, and the record fixed one early (def:pratt-order, convention of 11 September 2026): for strictly convex nondecreasing $c_1, c_2$ with $c_i(0)=0,$

The second equivalence is Pratt’s theorem, Pratt (1964), with the sign flipped: a convex cost is a concave utility read upside down, and $c''/c'$ is the Arrow–Pratt coefficient of absolute risk aversion. The convention that goes with the definition is the important part: every comparative static in the cost is stated along this order and only along it — for comparable pairs, never through a scalar summary such as an elasticity at a point or a top exponent, never across non-comparable costs. It is a partial order. Pure powers $e^\theta$ are totally ordered by $\theta;$ $e^2 e^{\beta e}$ by $\beta;$ $e^2 + a e^4$ by $a;$ $A(e^{ke}-1)$ by $k,$ with $A$ invisible since $c''/c' = k.$ The cosh family and the powers cross.

That last parenthesis — $A$ is invisible — is the whole problem, and the next section is its witness.

§ 2Why raw fails: it tracks units

The coefficient $c''/c'$ has dimension one over effort. Change the unit of effort and it changes; change the unit of payoff — multiply the cost by an amplitude $A$ — and it does not. The symmetric contest behaves in exactly the opposite way. By rem:sym-normalization the symmetric model at prize $v$ depends on $(A, c)$ only through $v/A$: the amplitude is a parameter of the game, the effort unit is not.

So take $c = A(\cosh e - 1)$ at two amplitudes. Both have $c''/c' = \coth e,$ so each is raw-comparable to the other both ways, and any monotone comparative static along $\succeq$ must give them the same value. But prop:cosh-floor-monotone makes the floor $m_c$ — the lowest Nash-live rung — a non-decreasing, unbounded staircase in $y = v/A,$ with $\sigma(1,1)$ live iff $y^3 - 4y \le 2.$ At $v = 1$ the amplitudes $A = 1$ and $A = \tfrac15$ sit on different steps.

This witness needs no mixed cost and lives outside X, since $c'(0) = 0$ holds here but the point is amplitude, not the origin. There is a harder one inside X at fixed amplitude: prop:pratt-sym-revival takes $c_s = (1+6s)e^2 + 18s\,e^4,$ strictly increasing in $\succeq$ as $s$ rises, and finds $\sigma(1,1)$ Nash at $s = 0,$ not Nash at $s = \tfrac16,$ Nash again at $s = 1.$ A collusive rung that dies and revives along a monotone path is not a comparative static that the order can state; clause (ii) of the symmetric floor conjecture was refuted in the raw order on 14 September 2026. Plate III returns to this path once the right order is in hand, and reads the pattern off exactly.

§ 3The normalization: pinned at both ends

The remark that diagnoses the failure also supplies the repair. Measure effort in reaches, $u = e/w$ with $w = c^{-1}(v),$ and payoff in prizes. This is a relabelling of the action set followed by a positive affine change of payoffs, so equilibria, the march and the dial correspond one to one, atoms scale by $1/w,$ masses are unchanged and $u_1 \mapsto u_1/v.$ The image model is the symmetric point with $\hat v = \hat w = 1$ and the cost

Every statement about the symmetric model is a statement about $\hat c$ alone. The reach-normalized Pratt order (def:reach-pratt-order, Sergio, 2 October 2026) is then the raw order applied to the normalized profiles:

The question that usually comes first — does this change the risk aversion without lifting the cost, with the reach held fixed? — has a precise yes. The reach is not frozen as a number; it is divided out, so every profile has reach 1 and prize 1. And “not lifted” is a theorem. Comparable profiles satisfy $\hat c_1 = \varphi\circ\hat c_2$ with $\varphi$ convex; both endpoints are pinned, so $\varphi(0) = 0$ and $\varphi(1) = 1;$ a convex function fixing $0$ and $1$ lies under the diagonal, $\varphi(x) \le x;$ hence

More convex means the chord bent downward between the same two anchors. The curve can sag; it cannot be raised, shifted or rescaled. In the raw order $\varphi(0) = 0$ is the only pin, and $c \mapsto Ac$ is “comparable both ways” precisely because nothing holds the far end.

The economics analogy is exact and worth saying once. Pratt’s order on utilities is affine-invariant for free, because cardinal utility is only defined up to $u \mapsto au + b$ and one may normalize $u(0) = 0,$ $u(W) = 1$ at no cost. Costs are not up to affine transformation — $A$ matters through $v/A$ — so the reach normalization is the step that restores to costs the invariance Pratt’s order has automatically for utilities.

§ 4The coefficient in the primitives

The primitives of the symmetric model are the prize $v$ and the cost $c;$ the reach $w = c^{-1}(v)$ is derived. By the chain rule,

with $\eta(e) = e\,c''(e)/c'(e)$ the elasticity of marginal cost. Two readings. In effort units the normalized coefficient is the raw coefficient times the reach — dimensionless, since $c''/c'$ is one over effort and $w$ is an effort. In elasticity form it is $\eta$ at the effort $wu,$ divided by $u;$ the $1/u$ is common to every cost, so comparing the normalized coefficients at the same $u$ is comparing $\eta_1(w_1 u)$ with $\eta_2(w_2 u),$ which is how the definition states it. The prize enters only through $w,$ that is, through where on the cost curve $\eta$ is read: a larger prize pushes the reach out and reads the elasticity further along $c.$

Is this relative risk aversion?

Yes. $\eta = e\,c''/c'$ is $-x\,u''(x)/u'(x)$ with the sign flipped and effort in the role of wealth. The dictionary:

And since $u\,\hat c''(u)/\hat c'(u) = \eta(wu),$ the relative coefficient of $\hat c$ at $u$ is the relative coefficient of $c$ at $wu$⁠: ordering absolute coefficients of the $\hat c$’s and ordering relative coefficients of the $\hat c$’s is one and the same order. So $\hat\succeq$ is relative risk aversion compared at equal fractions of the reach. What it adds to plain “relative” is the comparison point: $\eta$ is already invariant to $e \mapsto \lambda e,$ but the raw order compares two costs at a common absolute effort while each plays on its own reach; $\hat\succeq$ compares them at a common $e/w.$

Along the families the record uses: at $c = A(\cosh(ke)-1)$ the order is that of $W = kw$ with $\cosh W = 1 + v/A;$ for $e^\theta + \varepsilon e^2$ it is the order of the shape parameter $\rho = w^{\theta-2}/\varepsilon$ at fixed $\theta;$ for $e^2 + a e^4$ the profile is $(u^2 + \rho u^4)/(1+\rho)$ with $\rho = a w^2.$ Raising the prize walks each of these along its chain.

§ 5What the order has bought so far

One lemma, one reformulation, two families, and a witness explained.

The gate is the follower’s overshoot: against the leader at $w/2,$ player 2’s best deviation above $w/2$ is at $\tfrac34 w$ and pays $v - 2c(\tfrac34 w),$ so $\sigma(1,1)$ is Nash iff that is $\le 0.$ The comparison is the sag of §3, sharpened: $\hat c_1''/\hat c_1' \ge \hat c_2''/\hat c_2'$ makes $r = \hat c_1'/\hat c_2'$ nondecreasing; both derivatives integrate to $1$ over $[0,1],$ so either $r \equiv 1$ or $r \lt 1$ near $0$ and $r \gt 1$ near $1;$ hence $\hat c_1 - \hat c_2 = \int_0^{\cdot}(r-1)\,\hat c_2'$ falls and then rises from $0$ to $0,$ and is $\le 0$ throughout. In particular $\hat c_1(\tfrac34) \le \hat c_2(\tfrac34).$ The raw-order statement is false.

The revival, read correctly

Along $c_s = (1+6s)e^2 + 18s\,e^4$ at $v = 1$ the normalized profile is $(u^2 + \rho_s u^4)/(1+\rho_s)$ with $\rho_s = 18s\,w_s^2/(1+6s),$ and $\rho_s = 0, \tfrac12, \tfrac27$ at $s = 0, \tfrac16, 1$ — it rises and then falls back toward $0.$ The path climbs the raw order monotonically and is not a chain in $\hat\succeq.$ The gate reads

and the three values $0.5625,\ 0.4805,\ 0.5078$ fall on the right sides of $\tfrac12.$ Live, dead, live is what a non-monotone $\rho$ must produce; nothing is contradicted.

The conjecture, restated

conj:sym-convexity-floor clause (ii) — the floor is monotone in convexity at the symmetric point — now reads in $\hat\succeq$ (Sergio, 2 October 2026). In that order it is proved at $k = 1$ by the lemma above, and all three clauses hold along the two exactly solvable families, which are $\hat\succeq$-chains in $W = kw$⁠: cosh (prop:cosh-floor-monotone, the staircase in $y = v/A$) and exponential (prop:exp-sym-floor, the staircase affine in $V = 1 + v/A,$ outside X). Every measured GREEN chain — $e^\theta;$ $e^2 + ae^4$ with $\rho = aw^2$ increasing; $e^2 e^{\beta e},$ where $\eta = g(\beta wu)$ with $g$ increasing — is a $\hat\succeq$-chain. Clauses (i), (iii) and (ii) at $k \ge 2$ remain open on X.

§ 6Literature I — absolute versus relative

The organizing principle is old and sharp. Absolute risk aversion is the right coefficient when the risk is additive and of fixed size; relative risk aversion when the risk scales with wealth or with the stake, so that the problem is homogeneous. Both halves are in the founding papers.

Pratt (1964) defines both coefficients and proves both comparative statics in the same paper: for an additive gamble the risk premium is governed by $-u''/u';$ for a proportional gamble — wealth times $(1 + \tilde z)$ — the premium as a fraction of wealth is governed by $-x\,u''/u'.$ Arrow (1965), in the portfolio problem with one safe and one risky asset, gets two comparative statics from two coefficients: decreasing absolute risk aversion makes the dollar amount in the risky asset rise with wealth; increasing relative risk aversion makes the share fall, and constant relative risk aversion holds the share fixed. Whether the object is a level or a proportion decides the coefficient. Kihlstrom and Mirman (1974) is the caution that the order only extends cleanly across many commodities when the agents share ordinal preferences.

In the contest the object is a proportion. The remark of §3 says the symmetric game is homogeneous once effort is in reaches and payoff in prizes, and the principle then predicts that the relative coefficient, read in reach units, is the one that carries the comparative statics. The failure of $c''/c'$ in §2 is Arrow’s portfolio story run backwards: an absolute coefficient asked to sign a proportional comparative static.

§ 7Literature II — the “relative risk aversion against one” results

A recurring pattern in the applied theory: when a scale parameter of the risk shifts, the sign of the response is decided by whether relative risk aversion is above or below one. The mechanism is one line — $\tfrac{d}{dx}\bigl[x\,u'(x)\bigr] = u'(x)\bigl(1 - R(x)\bigr)$ — so $R \gtrless 1$ is the income-versus-substitution split, and it is where the CRRA case $R \equiv \gamma$ earns its tractability.

• Rothschild and Stiglitz (1971): a riskier return raises saving if $R \ge 1$ and nondecreasing, lowers it if $R \le 1$ and nonincreasing. Sandmo (1970) is the companion on saving under interest-rate risk.
• Fishburn and Porter (1976): a first-order improvement in the risky return raises risky investment for every such improvement iff $R \le 1.$ Cheng, Magill and Shafer (1987) and Hadar and Seo (1990) generalize the shift classes.
• Katz (1983), with the comment of Briys and Eeckhoudt (1985): a general class of problems in which the effect of a multiplicative shift is signed by relative, not absolute, risk aversion.
• Sandmo (1971), the competitive firm under price uncertainty: output responds to a fixed-cost change through decreasing absolute risk aversion, but to a multiplicative stretch of the price distribution through relative risk aversion. Stiglitz (1969): a proportional wealth tax is a multiplicative shock, so relative risk aversion signs the response of risk-taking.
• Insurance, from Mossin (1968) onward: with a loss that is a fixed fraction of wealth, coverage rises with wealth iff relative risk aversion is increasing.

The contest has its own version of “against one”: along the pure powers $\hat c = u^\theta,$ and the record’s thresholds in $\theta$ are thresholds in the constant relative coefficient. That is a remark, not a result; it is listed here because it is the shape the literature would expect.

§ 8Literature III — partial relative risk aversion, the stake-scaled coefficient

Menezes and Hanson (1970) and, independently, Zeckhauser and Keeler (1970) asked what coefficient governs a gamble whose size is set by a stake rather than by total wealth. With initial wealth $w_0$ and final wealth $w_0 + x(1 + \tilde z),$ $\tilde z$ zero-mean, the risk premium per unit of stake is governed by

Pratt’s absolute coefficient multiplied by the stake and evaluated at the wealth reached. It sits between the two Arrow–Pratt coefficients,

with the bookkeeping identity $R(w_0 + x) = P(x; w_0) + w_0\,A(w_0 + x).$ They prove that the premium per unit of stake is increasing, constant or decreasing in $x$ according as $P$ is, sort out the relations between the decreasing versions of $A,$ $R$ and $P$ (decreasing $A$ alone implies nothing about $P$), and the coefficient becomes the standard tool whenever a decision variable scales the risk while fixed wealth sits in the background — Briys and Eeckhoudt (1985) and Hadar and Seo (1990) use it to clean up the results of §7.

The paper is the precedent for the claim that “stake-scaled, read inside the stake” is the natural normalization — not relative to an arbitrary unit of effort, which is the raw order, and not relative to total wealth, which here is undefined.

§ 9Literature IV — when an order carries a comparative static

Having the right coefficient is half of it. The other half is knowing which comparative statics an Arrow–Pratt ordering can sign at all, and the literature has a precise theory of that.

Diamond and Stiglitz (1974) is the first systematic treatment of an optimal action as risk aversion rises, through mean-utility-preserving spreads; it is the standard reference for “more risk averse implies less of the risky action” in one-dimensional problems. The negative results come next. Kihlstrom, Romer and Williams (1981) and Ross (1981) show that with a background, uninsurable risk the Arrow–Pratt order does not imply the natural comparative static; Ross proposes the stronger order $u_2''/u_1'' \ge \lambda \ge u_2'/u_1',$ sharpened by Machina and Neilson (1987). Pratt and Zeckhauser (1987), Gollier and Pratt (1996) and Eeckhoudt, Gollier and Schlesinger (1996) characterize the utilities — proper, risk-vulnerable — for which background risk raises risk aversion.

The modern answer to “when does a comparative static in risk aversion survive” is Jewitt (1987): exactly when the relevant payoff difference single-crosses. That paper turned the question into a single-crossing problem, and Athey (2002) supplies the general machinery — log-supermodularity and single-crossing under uncertainty — of which Jewitt-type results are corollaries; Quah and Strulovici (2012) is the follow-up on aggregating single crossing. Gollier (1995) does the same clean-up for changes in risk rather than in risk aversion. Kimball (1990) adds the third-derivative coefficient, prudence $-u'''/u'',$ which governs precautionary motives in the way the second-derivative coefficient governs premia.

For a cost-side order the transferable tool is Jewitt–Athey: restate “more convex in $\hat\succeq$” as a single-crossing property of the relevant marginal condition, and the comparative static follows or fails exactly there. The lemma of §5 is a small instance — the gate $2c(\tfrac34 w) \ge c(w)$ is a single comparison of two normalized values, and the sag of §3 is the single-crossing fact that orders it.

§ 10Literature V — auctions and contests

Auctions

Holt (1980) and Riley and Samuelson (1981) establish the first comparative static of the field in risk aversion: with independent private values, risk aversion raises first-price bids, so first-price revenue exceeds second-price; the argument is Pratt’s $\varphi$-composition. Milgrom and Weber (1982) record the same ranking in the private-values case alongside the linkage principle. Maskin and Riley (1984) is the central paper: the optimal mechanism with risk-averse buyers screens on risk exposure, and the revenue-equivalence class collapses. Matthews (1983) solves the CARA optimal auction, and Matthews (1987) gives the clean comparative static by preference class: CARA buyers are indifferent between first and second price, DARA buyers prefer the second-price auction, IARA the first-price. That ranking is organized by the absolute coefficient because the relevant risk — win or lose a fixed surplus — is additive.

The relative coefficient appears where the problem is homogeneous. With $u(x) = x^{1-r}$ the first-price auction is homogeneous of degree one in values: scaling all values scales all bids, and the risk-averse equilibrium is the risk-neutral one with the number of bidders replaced by $1 + (n-1)/(1-r)$ — Cox, Smith and Walker (1988), used as the worked example in Riley–Samuelson. This is the same logic as the normalization of §3: under constant relative risk aversion the scale of the prize drops out and only the shape parameter remains. Eső and White (2004) introduce ex post risk about the object’s value: DARA bidders bid less and are better off, a prudence effect, the auction analogue of Kimball. Smith and Levin (1996) extend the format ranking; Fibich, Gavious and Sela (2006) treat all-pay auctions with risk-averse players; Hu, Matthews and Zou (2010) sign the optimal reserve in risk aversion — it falls in the first-price auction and is unaffected in the second; Che and Gale (2006) rank the formats beyond quasilinearity, with risk attitudes and wealth effects in one framework. Identification is Guerre, Perrigne and Vuong (2009) and Campo, Guerre, Perrigne and Vuong (2011).

Contests

The rent-seeking literature is closer to the contest here — effort is sunk, the prize is fixed — and it has no clean relative-coefficient result, which is itself informative: a Tullock payoff is not homogeneous in effort unless the cost is a pure power. Hillman and Katz (1984), Skaperdas and Gan (1995) and Konrad and Schlesinger (1997) find the effect of risk aversion on effort ambiguous in general and signed under CARA or limited-liability structures; Treich (2010) resolves the ambiguity with prudence; Cornes and Hartley (2012) handle heterogeneous risk aversion with share functions. None of them normalizes by a reach, because in a Tullock contest there is no effort at which the prize is exactly spent; the all-pay structure here is what makes $w = c^{-1}(v)$ a primitive of the game.

§ 11The record, and the open ends

What is not settled

• Two prizes. The remark that licenses the normalization is banked for the symmetric model. With $v_1 \ne v_2$ there are two reaches, and which normalization is the right one — the leader’s reach, the follower’s, or none — is not something the record decides. The Menezes–Hanson picture suggests the stake is whichever reach bounds the effort actually at risk, but that is a suggestion.
• Depth. The lemma is the $k = 1$ clause. Whether $\hat\succeq$ orders the floor at $k \ge 2$ on X is the open target of module 13; the families say yes, the general argument does not yet exist.
• Which comparative statics. By Jewitt’s criterion an order carries a comparative static only where the relevant difference single-crosses. The record has not asked, result by result, which of the cost comparative statics in the convention of §1 pass that test in $\hat\succeq.$

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