--- title: Truncated Cubic Contest Success Function url: https://www.emergentmind.com/topics/truncated-cubic-contest-success-function type: topic --- # Truncated Cubic Contest Success Function Searching arXiv for the cited papers and closely related contest-success-function work. The truncated cubic contest success function is a two-player contest success function in which the raw winning probability is specified by a symmetric cubic polynomial in efforts and then truncated to the unit interval. In the formulation studied in "Suppression and Empowerment in Contests" [2605.26639], the raw probability for player \(X\) against player \(Y\) is \[ P(x,y)=\frac{1}{2}+(x-y)\bigl(c-b(x+y)+axy\bigr), \] with \(x,y\ge 0\), \(c>0\), \(b>0\), and \(a\in\mathbb{R}\). The actual probability is \[ \bar P(x,y):=\min\{1,\max\{0,P(x,y)\}\}. \] Its distinctive feature is the strategic-feedback parameter \(a\), whose sign determines whether a lead in effort suppresses or empowers the opponent’s marginal effectiveness. The specification is also notable because it sits outside the logit-with-luck family axiomatized in "Contest success functions with luck" [2508.12934], except in degenerate monomial-plus-constant cases. ## 1. Formal specification The raw cubic CSF is \[ P(x,y)=\frac{1}{2}+(x-y)\bigl(c-b(x+y)+axy\bigr), \] where \(x\) and \(y\) are efforts, \(P(x,y)\) is the winning probability of \(X\), \(1-P(x,y)\) is the winning probability of \(Y\), \(c>0\) is the baseline marginal return to effort near zero, \(b>0\) is the own-effort concavity parameter, and \(a\in\mathbb{R}\) is the strategic-feedback parameter [2605.26639]. The specification is symmetric in the sense that \[ P(x,y)=1-P(y,x). \] The term “cubic” refers to the fact that \(P(x,y)\) is a polynomial of total degree three in the efforts. The cubic term is \((x-y)axy\). Observation 1 in [2605.26639] states that if a polynomial CSF satisfies \(P(x,y)=1-P(y,x)\) and has degree at most two, then \(\partial^2P/\partial x\,\partial y\equiv 0\). Cubic is therefore the minimal polynomial degree at which symmetry and a non-trivial cross-effect are compatible. Because the raw polynomial can leave the interval \([0,1]\), the literal contest uses truncation: \[ \bar P(x,y):=\min\{1,\max\{0,P(x,y)\}\}. \] The set on which truncation is inactive is the admissible domain \[ \mathcal D:=\{(x,y)\in\mathbb R_+^2:0y\), player \(X\) is leading. If \(a>0\), then \(P_{xy}>0\), so increasing the leader’s effort raises \(\partial P/\partial y\). Since player \(Y\)’s marginal gain from effort is \(-P_y\), it follows that \[ \frac{\partial}{\partial x}(-P_y)=-P_{xy}<0. \] Higher effort by the leader makes the trailing player’s own effort less effective. This is suppression. If \(x>y\) and \(a<0\), then \(P_{xy}<0\), hence \[ \frac{\partial}{\partial x}(-P_y)=-P_{xy}>0. \] Higher effort by the leader makes the trailing player’s effort more effective. This is empowerment. By symmetry, the same logic applies when \(Y\) is the leader. The parameter \(a\) is therefore a reduced-form institutional parameter governing how a local lead feeds back into the opponent’s marginal return. The paper organizes its comparative statics around three regimes: suppression for \(a>0\), empowerment for \(a<0\), and neutrality for \(a=0\) [2605.26639]. Under suppression, uncertainty and more information tend to reduce expected effort; under empowerment, they tend to increase expected effort; when \(a=0\), cross-effects vanish and best responses are independent. This feedback interpretation distinguishes the truncated cubic CSF from standard lottery contests. The generalized Tullock CSF, \[ P(x,y)=\frac{x^r}{x^r+y^r}, \] is suppressive as a functional-form property independent of parameter choice: \(P_{xy}(x,y)>0\) iff \(x>y\), so the leading player’s effort always reduces the trailing player’s marginal effectiveness [2605.26639]. ## 3. Complete-information equilibrium structure Under complete information, the model considers two players with effort spaces \(x,y\in\mathbb R_+\), prize normalized to \(1\), common cost parameter \(\theta>0\), and payoffs \[ U_1(x,y)=P(x,y)-\theta x,\qquad U_2(x,y)=1-P(x,y)-\theta y. \] For \(a\ne 0\), the analysis defines \[ \kappa=\frac{b}{a},\qquad \zeta=\kappa^2-\frac{c-\theta}{a} =\frac{b^2-a(c-\theta)}{a^2}. \] The symmetric first-order condition for a pure equilibrium is \[ ax^2-2bx+c-\theta=0, \] whose discriminant is \(4a^2\zeta\) [2605.26639]. Theorem 1 in [2605.26639] divides the unrestricted polynomial benchmark into a pure-equilibrium region and a mixed-equilibrium region. If \(\zeta\ge 0\), the unrestricted contest has a unique symmetric pure equilibrium, \[ x^*=y^*=\kappa-\mathrm{sgn}(a)\sqrt{\zeta}. \] If \(\zeta<0\), no pure equilibrium exists, and every unrestricted mixed equilibrium satisfies \[ \mathrm E[x]=\mathrm E[y]=\kappa,\qquad \mathrm{Var}(x)=\mathrm{Var}(y)=-\zeta. \] Conversely, any pair of distributions on \([0,\infty)\) with these moments constitutes an unrestricted mixed equilibrium. The mixed region arises only under sufficiently strong suppression. Under empowerment, \(a<0\) implies \(\zeta>0\), so equilibrium is always pure. The degenerate case \(a=0\) yields a dominant-strategy equilibrium, \[ x^*=y^*=\frac{c-\theta}{2b}>0. \] The pure-equilibrium formula extends continuously as \(a\to 0\) [2605.26639]. The best-response geometry clarifies the difference between regimes. The interior best response can be written as \[ x^{BR}(y)=\frac{y+\kappa}{2}+\frac{\zeta}{2(y-\kappa)}, \] with second derivative \[ \frac{d^2x^{BR}(y)}{dy^2}=\frac{\zeta}{(y-\kappa)^3}. \] Under empowerment, the interior branch is always convex. Under suppression, it is concave when \(\zeta>0\) and convex when \(\zeta<0\). The curvature flip occurs exactly at \(\zeta=0\), the transition from pure to mixed equilibrium [2605.26639]. Equilibrium effort is single-peaked in the feedback parameter. Theorem 2 states that on the pure-equilibrium region, equilibrium effort \(x^*(a)\) is increasing in \(a\), whereas on the mixed-equilibrium region every mixed equilibrium has \(\mathrm E[x]=b/a\), which is decreasing in \(a\). Expected equilibrium effort is maximized at the boundary \[ b^2=a(c-\theta). \] Total effort is likewise single-peaked in \(a\) [2605.26639]. In the literal truncated contest, equilibrium survival requires additional restrictions. For pure equilibrium, Proposition 3 states that the symmetric pure equilibrium is the unique on-domain equilibrium iff its expected utility is nonnegative, that is, \[ \frac{1}{2}\ge \theta x^*. \] Otherwise players deviate to zero effort. For mixed equilibrium, on-domain equilibria must satisfy the unrestricted moment conditions together with participation, \[ \frac{1}{2}\ge \theta\kappa. \] Existence is then support-dependent. The paper provides branch-specific conditions for canonical two-point families, primitive sufficient conditions ensuring a nonempty mixed region in the truncated contest, and a finite threshold \(\bar a_F\) beyond which no on-domain mixed equilibrium exists [2605.26639]. ## 4. Incomplete information and information design The incomplete-information extension assumes IID private costs \(\theta_i\) with support \([\alpha,\beta]\subset(0,c)\). The distribution \(F\) may have atoms. Let \[ M_1=\mathrm E[\theta],\qquad M_2=\mathrm E[\theta^2],\qquad \sigma_\theta^2=M_2-M_1^2. \] The paper defines \[ \Delta=c-M_1,\qquad \kappa=\frac{b}{a},\qquad \omega=\frac{\sigma_\theta^2}{a^2},\qquad \zeta=\kappa^2-\frac{\Delta}{a} =\frac{b^2-a\Delta}{a^2}. \] In the affine relaxation, where actions may be any real number, Theorem 3 gives a unique symmetric Bayesian Nash equilibrium of the form \[ x(\theta)=k\theta+d,\qquad k<0, \] so effort is strictly decreasing in type [2605.26639]. A central structural property is that the equilibrium depends on the type distribution only through its first two moments. The equilibrium is in closed form, and the variance of equilibrium effort is also explicit. When \(a=0\), each type has a dominant strategy \[ x(\theta)=\frac{c-\theta}{2b}>0, \] and the affine formulas converge smoothly to this case. As uncertainty vanishes, the Bayesian equilibrium selects the complete-information benchmark. If \(\sigma_\theta^2\to 0\) and \(\zeta\ge 0\), expected effort converges to the pure equilibrium and the variance of effort converges to zero. If \(\zeta<0\), expected effort converges to \(\kappa\) and effort variance converges to \(-\zeta\), which is exactly the mixed-equilibrium moment profile [2605.26639]. This provides a natural selection result among the many complete-information mixed equilibria. Imposing nonnegativity yields symmetric Bayesian equilibria of cutoff-affine form, \[ x(\theta)=\lambda(t-\theta)_+,\qquad \lambda>0,\ t\ge \alpha. \] If \(t\ge \beta\), all types are active and the equilibrium coincides with the affine benchmark. If \(t<\beta\), higher-cost types choose zero and lower-cost types use a linear rule. Under empowerment, one can show that \(t\ge \beta\) always holds, so dropout never occurs. Under suppression, sufficiently strong feedback generates dropout, and for atomless priors the dropout cutoff is strictly decreasing in \(a\), while the dropout rate is strictly increasing in \(a\) [2605.26639]. The same sign asymmetry governs information disclosure. In the symmetric IPV persuasion framework following Bergemann et al. 2022, ex ante expected effort depends on the information structure only through the variance of posterior mean costs. Theorem 5 implies that an effort-maximizing designer chooses no disclosure under suppression, full disclosure under empowerment, and is indifferent when \(a=0\) [2605.26639]. This all-or-nothing disclosure rule is one of the paper’s sharpest implications. ## 5. Relation to axiomatized contest families The truncated cubic CSF is not merely a variant of the Tullock or logit-with-luck families. The paper "Contest success functions with luck" [2508.12934] studies CSFs on finite contestant sets under Strict Monotonicity, Luce’s Choice Axiom, and Homogeneous Relative Externality. Under Strict Monotonicity and Luce’s Choice Axiom, every CSF has logit form \[ p_i^M(x^M)=\frac{f_i(x_i)}{\sum_{j\in M}f_j(x_j)}, \] with strictly increasing impact functions \(f_i\). Adding Homogeneous Relative Externality forces \[ f_i(x_i)=b_i+a_i x_i^r, \] with a common exponent \(r>0\) across contestants, \(a_i>0\), and \(b_i\ge 0\) [2508.12934]. Within that axiomatization, the only admissible “cubic” impact functions are single-power forms such as \(f(x)=ax^3\) or \(f(x)=d+ax^3\). A generic cubic polynomial, \[ f(x)=ax^3+bx^2+cx+d, \] is incompatible with Homogeneous Relative Externality unless all but one power coefficient are zero. The same analysis states that pointwise truncation, bounded-domain truncation, or piecewise polynomial truncation generally destroy the required homogeneity structure [2508.12934]. This places the truncated cubic CSF of [2605.26639] in sharp conceptual contrast with the axiomatized class of [2508.12934]. The former is a two-player probability function defined directly as a cubic polynomial in both players’ efforts and then clipped to \([0,1]\); the latter characterizes ratio-form CSFs whose admissible impact functions are affine in a monomial. A plausible implication is that the truncated cubic benchmark should be understood as an alternative reduced form rather than as a member of the HRE-based logit family. The comparison also clarifies what is special about standard lottery contests. In [2605.26639], the generalized Tullock CSF is suppressive by construction. In [2508.12934], the Tullock form appears when homogeneity replaces the weaker Homogeneous Relative Externality and thereby rules out luck. The two papers thus approach contest technology from different directions: one enlarges the Tullock family by allowing additive constants in impact functions, while the other departs from ratio forms altogether in order to model suppression and empowerment symmetrically. ## 6. Interpretation, uses, and limitations The economic interpretation of the strategic-feedback parameter is institutional rather than purely mechanical. The paper associates suppression with environments in which a lead reduces the rival’s marginal effectiveness, and empowerment with environments in which a lead raises it. Examples include R&D races under strong versus weak appropriability, political competition under capture-prone versus independent media, capital and talent markets under different flow patterns, and platforms under proprietary lock-in versus interoperability and data portability [2605.26639]. The truncated cubic CSF is useful because it is deliberately minimal. It is the lowest-degree symmetric polynomial that allows a cross-effect between efforts, avoids the singularity at zero effort that plagues lottery CSFs, and yields closed-form mixed equilibria under complete information together with a unique affine Bayesian Nash equilibrium under arbitrary IID type distributions [2605.26639]. These features make it a tractable benchmark for information design, mixing, and comparative statics. At the same time, the framework has explicit limitations. The model is static; \(a\) is a property of the contest technology or institution rather than a dynamic race effect. The parameter \(a\) is a one-dimensional reduction of a higher-dimensional institutional structure. Because truncation is essential, the raw polynomial is meaningful chiefly when equilibrium play remains inside the admissible domain \(\mathcal D\). Under extreme suppression, clipping and dropout can move the game outside the range in which the cubic benchmark is a good approximation [2605.26639]. Several familiar conclusions of contest theory are therefore regime-specific rather than universal. The claim that uncertainty tends to reduce effort, or that more information tends to lower effort, matches the cubic model only under suppression. Under empowerment, the sign reverses. This suggests that many standard conclusions associated with lottery contests reflect suppressive benchmarks rather than contests as such [2605.26639]. Source: https://www.emergentmind.com/topics/truncated-cubic-contest-success-function