Fractional Pseudo-Games

• Fractional pseudo-games are fractional augmentations in paired comparisons that add symmetric wins and losses to every competitor pair, stabilizing the likelihood estimation.
• They effectively shrink pairwise ability differences by incorporating a penalized likelihood, yielding finite estimates even in sparse or separated data scenarios.
• The method provides an intuitive augmented-data interpretation with direct links to ridge regularization, applicable in both Bradley–Terry and Thurstone–Mosteller models.

Searching arXiv for the cited papers to ground the article in current literature. In the most explicit current usage, fractional pseudo-games are artificial, fractional wins and losses added symmetrically between every pair of competitors in a paired comparison model. In a linear paired comparison model with latent abilities $\theta_1,\dots,\theta_J$, they augment the data as if every unordered pair had played an additional balanced “pseudo-matchup” with total weight $2\delta$, where $\delta\ge 0$ may be noninteger. This produces a penalized likelihood with an intuitive augmented-data interpretation, yields finite, shrunken estimates when the ordinary maximum likelihood estimator is unstable or fails, and applies in both Bradley–Terry and Thurstone–Mosteller models (Glickman, 2 Jun 2026). In other literatures, the phrase also appears in a broader, interpretive sense for models with fractional utilities, pseudo-polynomial dynamic programming, or nonlocal game representations; those extensions are analogical rather than a single unified formalism.

1. Formal construction in paired comparison models

The paired comparison setup considered in the regularization literature has $J$ competitors with latent abilities $\theta_1,\dots,\theta_J$. In a linear paired comparison model, when $i$ plays $j$, the win probability for $i$ is

$p_{ij}=\Pr(Y_{ij}=1)=F(\theta_i-\theta_j),$

with $F$ monotone and satisfying $2\delta$0. Two key choices are Bradley–Terry, with $2\delta$1, and Thurstone–Mosteller, with $2\delta$2. If $2\delta$3 is the number of times $2\delta$4 defeats $2\delta$5, $2\delta$6, and $2\delta$7, then the ordinary log-likelihood is

$2\delta$8

Fractional pseudo-games replace the observed counts by augmented counts

$2\delta$9

for all $\delta\ge 0$0, where $\delta\ge 0$1 may be noninteger. The construction is symmetric: for each unordered pair $\delta\ge 0$2, one adds $\delta\ge 0$3 fractional wins for $\delta\ge 0$4 against $\delta\ge 0$5 and $\delta\ge 0$6 fractional wins for $\delta\ge 0$7 against $\delta\ge 0$8. The augmented log-likelihood becomes

$\delta\ge 0$9

The substantive interpretation is direct: beyond the observed matches, the model behaves as if there were an artificial mini-tournament in which every pair of competitors played a tiny symmetric series producing equal wins and losses (Glickman, 2 Jun 2026).

Because the construction depends only on the link function $J$0, it is identical in Bradley–Terry and Thurstone–Mosteller models except for the choice of $J$1. This makes fractional pseudo-games a model-level augmentation rather than a device specific to one link.

2. Penalized-likelihood structure and local ridge equivalence

The augmented likelihood can be written as

$J$2

Since $J$3, each summand is maximized when $J$4. Thus the pseudo-game penalty shrinks all pairwise probabilities toward $J$5, equivalently all differences $J$6 toward $J$7. In this sense, fractional pseudo-games are a penalized-likelihood device whose penalty acts on relative strengths rather than on absolute levels (Glickman, 2 Jun 2026).

For the Bradley–Terry model, with $J$8,

$J$9

so

$\theta_1,\dots,\theta_J$0

The induced penalty depends only on pairwise differences, is maximized at $\theta_1,\dots,\theta_J$1 for all pairs, and for large $\theta_1,\dots,\theta_J$2 grows roughly linearly in $\theta_1,\dots,\theta_J$3 rather than quadratically. This contrasts with ridge regularization,

$\theta_1,\dots,\theta_J$4

which penalizes individual coefficients.

The relation to ridge is nevertheless explicit. A Taylor expansion near $\theta_1,\dots,\theta_J$5 gives

$\theta_1,\dots,\theta_J$6

Hence, ignoring constants and higher-order terms,

$\theta_1,\dots,\theta_J$7

Using

$\theta_1,\dots,\theta_J$8

and imposing the usual centering constraint $\theta_1,\dots,\theta_J$9, one obtains the local equivalence

$i$0

Accordingly, for small pairwise differences or modest regularization, fractional pseudo-games behave like ridge regularization with strength $i$1, while retaining a likelihood interpretation in terms of augmented game counts (Glickman, 2 Jun 2026).

3. Connectivity, separation, and identifiability

In ordinary Bradley–Terry and Thurstone–Mosteller models, finite maximum likelihood estimates exist only when the comparison graph satisfies Ford’s strong connectivity condition: every nonempty subset of competitors both beats and is beaten by competitors outside the subset. If the graph is disconnected, or if one group always beats another group, the likelihood is maximized on the boundary and some ability differences diverge to $i$2. Fractional pseudo-games alter this geometry by ensuring that every pair $i$3 has at least $i$4 wins for $i$5 and $i$6 wins for $i$7, so the augmented comparison graph is complete and the pseudo part is perfectly balanced (Glickman, 2 Jun 2026).

The effect on estimation is twofold. First, no pair can remain entirely one-sided in the augmented data. Second, infinite ability gaps cease to be optimal, because sending $i$8 to $i$9 or $j$0 drives $j$1 to $j$2. The pseudo-game penalty therefore pushes separated or nearly separated probabilities back toward $j$3. This removes the separation induced by zeros in the contingency table and yields a finite maximizer of the penalized likelihood, subject to standard technical conditions on $j$4 (Glickman, 2 Jun 2026).

That stabilization does not, however, resolve the usual location indeterminacy. Because both the ordinary likelihood and the pseudo-game penalty depend only on differences $j$5, replacing each $j$6 by $j$7 leaves the objective unchanged. A linear constraint such as

$j$8

is still required. Fractional pseudo-games therefore regularize pairwise gaps and connectivity but do not anchor the overall level of the ability scale.

4. Phantom-player augmentation and its contrast with pseudo-games

The same paper develops a second augmentation based on a phantom player. One introduces an artificial competitor $j$9 with fixed ability $i$0, and each real competitor $i$1 receives one pseudo-win and one pseudo-loss against that player, each with case weight $i$2. The augmented log-likelihood is

$i$3

For Bradley–Terry this becomes

$i$4

Unlike pseudo-games, this penalty acts on individual $i$5 values and anchors the scale to $i$6. Expanding around $i$7 yields the local ridge correspondence $i$8, again with logistic rather than quadratic tails (Glickman, 2 Jun 2026).

The contrast between the two augmentations is structural rather than merely notational.

Feature	Fractional pseudo-games	Phantom player
Where shrinkage acts	Pairwise differences $i$9	Individual $p_{ij}=\Pr(Y_{ij}=1)=F(\theta_i-\theta_j),$0
Identifiability	Still needs $p_{ij}=\Pr(Y_{ij}=1)=F(\theta_i-\theta_j),$1	Anchored by $p_{ij}=\Pr(Y_{ij}=1)=F(\theta_i-\theta_j),$2
Local ridge equivalence	$p_{ij}=\Pr(Y_{ij}=1)=F(\theta_i-\theta_j),$3 under centering	$p_{ij}=\Pr(Y_{ij}=1)=F(\theta_i-\theta_j),$4
Data interpretation	$p_{ij}=\Pr(Y_{ij}=1)=F(\theta_i-\theta_j),$5 extra wins and $p_{ij}=\Pr(Y_{ij}=1)=F(\theta_i-\theta_j),$6 extra losses for every pair	$p_{ij}=\Pr(Y_{ij}=1)=F(\theta_i-\theta_j),$7 wins and $p_{ij}=\Pr(Y_{ij}=1)=F(\theta_i-\theta_j),$8 losses versus a fixed-strength reference

This comparison clarifies a common misconception. Fractional pseudo-games and phantom-player regularization are both augmented-data constructions, but they are not interchangeable at the level of identifiability: pseudo-games preserve location nonidentifiability, whereas the phantom player eliminates it directly (Glickman, 2 Jun 2026).

5. Tuning, predictive calibration, and the 2025 MLB illustration

A distinctive feature of fractional pseudo-games is that the tuning parameter $p_{ij}=\Pr(Y_{ij}=1)=F(\theta_i-\theta_j),$9 admits a direct predictive interpretation. In the minimal setting where players A and B have played one actual game and A won, with no other data, the pseudo-game augmentation produces augmented counts $F$0 for A over B and $F$1 for B over A. The resulting estimate is

$F$2

If one wishes this minimal-data estimate to equal a target probability $F$3, then

$F$4

The paper gives two examples. If one win should imply $F$5, then $F$6, corresponding to very weak shrinkage. If one win should imply $F$7, then $F$8 is larger, implying stronger regularization. This yields a subject-matter tuning rule: decide how convincing a single unopposed win ought to be, then back out $F$9 (Glickman, 2 Jun 2026).

A second tuning route is cross-validation. The procedure described is: choose a grid of candidate $2\delta$00 values, use $2\delta$01-fold cross-validation, fit the pseudo-game model on the training folds for each $2\delta$02, evaluate the ordinary Bradley–Terry log-likelihood on the held-out fold, sum validation log-likelihoods over folds, and select the $2\delta$03 that maximizes this criterion. The same framework applies to ridge and phantom-player tuning (Glickman, 2 Jun 2026).

The empirical illustration uses the 2025 Major League Baseball regular season, comprising 30 teams and 2,430 games. Four Bradley–Terry models are fit: ordinary Bradley–Terry, ridge-penalized Bradley–Terry with optimal $2\delta$04, pseudo-game Bradley–Terry with optimal $2\delta$05, and phantom-player Bradley–Terry with optimal $2\delta$06, all tuned by 10-fold cross-validation. The pseudo-game and phantom-player estimates are substantially shrunk relative to the ordinary fit, especially for the most extreme teams. For the Milwaukee Brewers versus Colorado Rockies strength difference, the fitted differences are 1.365 for ordinary Bradley–Terry, 0.820 for ridge, 0.907 for pseudo-games, and 0.887 for the phantom player. Under Bradley–Terry, a difference of 1.365 implies about 0.797 win probability, whereas a difference of 0.887 implies about 0.708. The paper reports that both pseudo-game and phantom-player regularization approximate ridge closely, with the phantom construction slightly closer pointwise in the scatter plots. It also interprets $2\delta$07 as giving each pair of teams more than one pseudo-win and more than one pseudo-loss, while $2\delta$08 means each team effectively plays 80 pseudo-games against a fixed-strength opponent (Glickman, 2 Jun 2026).

6. Bayesian flavor and regularization context

The augmented log-likelihood

$2\delta$09

has an immediate prior-like interpretation. Exponentiation yields

$2\delta$10

so the pseudo-game scheme corresponds to a pairwise factor

$2\delta$11

combined with a centering constraint for identifiability. For Bradley–Terry this becomes a product of logistic-shaped factors in the pairwise differences; for Thurstone–Mosteller it becomes a product of probit-shaped factors. The phantom-player augmentation analogously induces

$2\delta$12

The paper therefore presents pseudo-games and phantom players as either pseudo-data constructions or specific shrinkage priors, with links to power priors and Jeffreys-type penalties in generalized linear models (Glickman, 2 Jun 2026).

Within the broader regularization literature for paired comparisons, the comparison points named are ridge regularization, bias reduction and Jeffreys priors, and Bayesian hierarchical models. Ridge provides a quadratic penalty that stabilizes estimation and handles disconnected graphs by shrinkage; bias reduction and Jeffreys-prior methods yield adjusted scores and implicit pseudo-data that control separation; Bayesian hierarchical models place explicit priors on strengths and provide full posterior uncertainty. Fractional pseudo-games differ in emphasis because they augment the comparison graph itself by adding balanced pseudo-results between every pair and encode the prior belief that, absent strong evidence, every matchup is approximately $2\delta$13–$2\delta$14 (Glickman, 2 Jun 2026).

7. Broader and analogical uses across adjacent literatures

Outside paired comparison modeling, the phrase fractional pseudo-games is used more loosely. In the literature on fractional hedonic games on tree-like graphs, the paper on welfare maximization states that it does not itself use the term, but that its content “suggests a natural interpretation”: games with fractional utilities and pseudo-polynomial dynamic programming on bounded-treewidth or block-graph structures (Hanaka et al., 2023). There the relevant fractional structure is coalition utility

$2\delta$15

and the salient algorithmic point is that utilitarian and egalitarian welfare on tree-like graphs admit $2\delta$16 dynamic programs, while utilitarian welfare on block graphs is solvable in $2\delta$17 time (Hanaka et al., 2023). This suggests an algorithmic reading of the phrase, centered on normalized coalition payoffs and pseudo-polynomial state tracking.

A related but distinct line studies strategyproof mechanisms for additively separable hedonic games and fractional hedonic games. There, fractional hedonic games are presented as a canonical model in which coalition utility is the average

$2\delta$18

and the paper explicitly remarks that this difference from additive utilities is crucial for incentives and welfare guarantees (Flammini et al., 2017). The mechanism-design results include the statement that for general valuation functions there is no randomized strategyproof acceptable mechanism with bounded approximation ratio for either additively separable or fractional hedonic games, and that for fractional hedonic games with simple valuations a deterministic, polynomial-time, strategyproof 2-approximation based on maximum weighted matching is tight (Flammini et al., 2017). In this literature, “fractional pseudo-games” functions as an interpretive umbrella for coalition games with averaged rather than summed payoffs.

The nonlocal PDE literature supplies a different analogical usage. “Non-Local Tug-of-War and the Infinity Fractional Laplacian” constructs a “fractional tug-of-war” in which the move is determined by a symmetric $2\delta$19-stable Lévy process rather than a local coin flip, and the exposition describes it as a prototypical example of what one might call a fractional pseudo-game (Bjorland et al., 2010). The limiting value function solves a deterministic nonlocal integro-differential equation involving the infinity fractional Laplacian, and the game interpretation extends to both Dirichlet and double-obstacle problems (Bjorland et al., 2010). Here the phrase denotes a stochastic control game with nonlocal, heavy-tailed jumps rather than an augmented-data device.

Network-game theory provides yet another usage. “Preference Games and Personalized Equilibria, with Applications to Fractional BGP” frames preference games, fractional BGP, fractional BBC, and personalized equilibria as games in which players split effort, flow, or budget fractionally and payoffs are determined by personalized matchings or flows rather than a single global joint distribution (0812.0598). The paper proves that there are no fully polynomial-time approximation schemes for computing equilibria in fractional BGP and fractional BBC games unless PPAD is in FP, and it introduces personalized equilibria as a flow-based equilibrium notion for matrix games, with polynomial-time computation for the two-player case and PPAD-hardness of approximation for $2\delta$20 (0812.0598). In this setting, the phrase points to fractionalized strategies and player-specific couplings.

Modified fractional hedonic games sharpen the point that small changes in normalization can materially alter stability theory. In that model, utility is averaged over the other coalition members,

$2\delta$21

and the paper explicitly describes the model as a particular instance of broader fractional coalition games, or “fractional pseudo-games,” with utility of the form $2\delta$22 (Monaco et al., 2018). The paper proves, among other results, that in unweighted graphs one can compute a socially optimal strong Nash equilibrium in polynomial time, while for general weighted graphs one can compute a core-stable outcome within a factor $2\delta$23 of optimal social welfare (Monaco et al., 2018).

Taken together, these literatures indicate that fractional pseudo-games has a precise technical meaning in paired comparison regularization, but also a wider analogical role across coalition formation, network equilibria, and nonlocal game representations. The common thread is a replacement of purely observed or purely integral interactions by fractional, auxiliary, or personalized structures that preserve a game-like interpretation while changing the analytical object being optimized.