State-Conditioned Empirical Win Rate

• State-conditioned empirical win rate is a method that assigns success probabilities based on formally defined states in systems like gaming and UI evaluation.
• The approach integrates dynamic programming, empirical frequency estimation, and Bayesian smoothing to accurately predict outcomes even in sparse-state regimes.
• Applications span diverse domains, from game strategy (e.g., Fire Emblem) to sports analytics and interface performance, illustrating its broad practical significance.

State-conditioned empirical win rate is a conditional success functional defined on a state space and evaluated either as an exact probability, an empirical frequency, or a posterior predictive quantity. In Brockmann’s Fire Emblem arena formulation, it is $P(m,n)$, the probability that the player ultimately wins from state $(m,n)$ (Brockmann, 2018). Closely related constructions appear in transition-level recommendation, where win-rate change is conditioned on strategy state, behavioral subtype, and current win-rate bucket (Heo et al., 21 May 2026); in GUI evaluation, where the principal “Overall Win Rate” is exact target-state attainment after a single interaction (Ji et al., 30 Apr 2026); and in soccer, where posterior predictive win probabilities are conditioned on a ten-feature game-state vector over rescaled match time (Robberechts et al., 2019). Across these settings, the central idea is the same: success is not summarized globally, but conditioned on a formally specified state that is intended to be sufficient for prediction or control.

1. Formal object and semantic scope

The most general expression in the supplied formulations is

$$W_s=\Pr[\text{win}\mid s]=\sum_{s'}\Pr[s\to s']\,W_{s'}.$$

This writes the win-rate function as a state-indexed fixed point over one-step transitions. In deterministic evaluation pipelines the same object may appear as an average of indicators rather than as a recurrence, but the conditioning principle is unchanged: one asks for success probability or success frequency given the current state.

The Fire Emblem arena analysis makes this explicit by defining

$$P(m,n)=\text{probability that the player ultimately wins from state }(m,n),$$

where $m$ and $n$ encode the remaining number of standard hits required to kill the player and the enemy, respectively (Brockmann, 2018). The Clash Royale TQP framework instead conditions on a strategy state $s$, a behavioral subtype $u$, and a win-rate bucket $b$, then defines empirical quantities such as $\mathrm{stay\_baseline}(s,u,b)$ and the net transition effect $(m,n)$0 (Heo et al., 21 May 2026). FineState-Bench uses exact-state attainment as the relevant notion of “win,” with

$(m,n)$1

that is, Exact State Success Rate at Interact (Ji et al., 30 Apr 2026). Robberechts et al. similarly define in-game win probability in soccer as a state-conditioned posterior predictive over future goals rather than as a raw count ratio (Robberechts et al., 2019).

These examples indicate that “win rate” need not mean only eventual binary victory in a game. It can denote eventual battle victory, incremental win-rate improvement after a strategic switch, or exact goal-state attainment in a benchmarked interaction. The common requirement is a formally specified state and a success variable evaluated conditionally on that state.

2. State specification and conditioning variables

The quality of a state-conditioned empirical win rate depends on how the state is constructed. In the Fire Emblem recurrence, the state is intentionally small: $(m,n)$2 with fixed parameters $(m,n)$3 and booleans $(m,n)$4 indicating whether the player or enemy is fast enough to follow up (Brockmann, 2018). This representation compresses health and combat rules into a low-dimensional sufficient summary for the recurrence.

In Clash Royale, state is not purely mechanical. The authors cluster each deck into one of $(m,n)$5 archetypes, which they call strategy states,

$(m,n)$6

and further assign players to one of three subtypes via k-means on overall_switch_rate, loss_reactivity, and avg_change_magnitude. The resulting subtypes are $(m,n)$7 One-deck Loyalist, $(m,n)$8 Loss-Reactive Switcher, and $(m,n)$9 Flex Player. Conditioning is then carried out jointly on $$W_s=\Pr[\text{win}\mid s]=\sum_{s'}\Pr[s\to s']\,W_{s'}.$$0, where $$W_s=\Pr[\text{win}\mid s]=\sum_{s'}\Pr[s\to s']\,W_{s'}.$$1 is the current win-rate bucket (Heo et al., 21 May 2026). This suggests a broader interpretation in which state includes both environment configuration and persistent behavioral phenotype.

FineState-Bench adopts an instance-level state description

$$W_s=\Pr[\text{win}\mid s]=\sum_{s'}\Pr[s\to s']\,W_{s'}.$$2

where $$W_s=\Pr[\text{win}\mid s]=\sum_{s'}\Pr[s\to s']\,W_{s'}.$$3 is the screenshot, $$W_s=\Pr[\text{win}\mid s]=\sum_{s'}\Pr[s\to s']\,W_{s'}.$$4 is the target control, $$W_s=\Pr[\text{win}\mid s]=\sum_{s'}\Pr[s\to s']\,W_{s'}.$$5 is the exact, quantifiable goal state, and $$W_s=\Pr[\text{win}\mid s]=\sum_{s'}\Pr[s\to s']\,W_{s'}.$$6 and $$W_s=\Pr[\text{win}\mid s]=\sum_{s'}\Pr[s\to s']\,W_{s'}.$$7 are the localization box and interactable core, respectively (Ji et al., 30 Apr 2026). Here the conditioning variables are not latent features learned from logs; they are explicit annotations designed to guarantee exact-state evaluation.

In soccer, the state at time $$W_s=\Pr[\text{win}\mid s]=\sum_{s'}\Pr[s\to s']\,W_{s'}.$$8 is a ten-dimensional feature vector $$W_s=\Pr[\text{win}\mid s]=\sum_{s'}\Pr[s\to s']\,W_{s'}.$$9, computed in one of $$P(m,n)=\text{probability that the player ultimately wins from state }(m,n),$$0 equally spaced time frames after rescaling the $$P(m,n)=\text{probability that the player ultimately wins from state }(m,n),$$1 minutes of play. The features include Game Time, Score Differential, Rating Differential, Team Goals so far, Reds, Yellows, Goal-scoring Opportunities per frame, Attacking Passes per frame, Expected Threat per frame, and Duel Strength in the last 10 frames (Robberechts et al., 2019). This is a richer contextual state, intended to support stable probabilistic inference in a low-scoring sport.

A plausible implication is that state-conditioned empirical win rate is less a single metric than a family of conditional estimands whose meaning is determined by the granularity and sufficiency of the state representation.

3. Exact recurrence and dynamic-programming computation

The most explicit exact computation appears in the Fire Emblem arena model. In the simplest case, with no criticals and no follow-ups, the recurrence is

$$P(m,n)=\text{probability that the player ultimately wins from state }(m,n),$$2

which rearranges to

$$P(m,n)=\text{probability that the player ultimately wins from state }(m,n),$$3

When critical hits are added but follow-ups are still absent, a strike is miss, regular hit, or critical hit, and one round yields $$P(m,n)=\text{probability that the player ultimately wins from state }(m,n),$$4 outcomes; the resulting recurrence shifts $$P(m,n)=\text{probability that the player ultimately wins from state }(m,n),$$5 and $$P(m,n)=\text{probability that the player ultimately wins from state }(m,n),$$6 by $$P(m,n)=\text{probability that the player ultimately wins from state }(m,n),$$7, $$P(m,n)=\text{probability that the player ultimately wins from state }(m,n),$$8, or $$P(m,n)=\text{probability that the player ultimately wins from state }(m,n),$$9 depending on whether the strike misses, hits normally, or crits (Brockmann, 2018).

Boundary conditions determine the semantics of terminal states: $m$0 The simultaneous lethal case is resolved in the no-follow-up setting by setting $m$1, since the player always strikes first in a round; when follow-ups alter the order, the formulation requires an indicator function $m$2 to zero out impossible post-death strikes (Brockmann, 2018).

Once the recurrence is written in the generic form

$m$3

the entire table can be filled by dynamic programming in $m$4 time. The supplied implementation recipe allocates a two-dimensional array, fills boundary cells, iterates over $m$5 and $m$6, substitutes already computed neighbors, and divides by $m$7 where necessary. The summary emphasizes $m$8 time, $m$9 memory, optional reduction to rolling rows, clamping of negative indices from $n$0 and $n$1, optional exact rational arithmetic or high precision, and precomputation of denominators (Brockmann, 2018).

The same source generalizes the pattern to any two-player, turn-based, finite-state game with probabilistic moves: define a finite state $n$2, enumerate immediate stochastic outcomes $n$3, assign boundary values on terminal states, topologically sort $n$4, and fill a table of size $n$5 in $n$6 time, where $n$7 is average fan-out (Brockmann, 2018). This is the canonical exact form of a state-conditioned win-rate computation.

4. Empirical transition systems and baseline-adjusted win rates

An exact recurrence becomes an empirical estimator when transition probabilities are replaced by observed frequencies. For observed arena-battle paths, the supplied Fire Emblem summary defines empirical transition frequencies

$n$8

as the fraction of visits to state $n$9 that next go to $s$0, and then defines $s$1 as the solution of

$s$2

with the same boundary conditions. In practice, one may replace $s$3 by their sample estimates and compute the resulting DP; under mild mixing assumptions, $s$4 converges to the true win-frequency for large datasets (Brockmann, 2018).

The Clash Royale TQP framework adds an important refinement: empirical win-rate change is not interpreted naively, because players on a losing streak tend to recover their win rate even if they do not switch decks. To control for this, the authors define the state-matched baseline

$s$5

where

$s$6

The TQP model predicts the raw transition target

$s$7

using a GRU-based encoder plus MLP head trained to minimize MAE, and then computes the net transition effect

$s$8

A positive net effect means the switch outperformed simply staying under identical conditions (Heo et al., 21 May 2026).

This baseline adjustment is tied to the paper’s critique of the Zero Switching Cost Assumption. By conditioning the baseline on the same subtype $s$9 and state $u$0, the framework is described as automatically accounting for each group’s characteristic mastery-loss curve and tilt-induced volatility; accordingly, the net effect measures the extra gain or loss due to changing decks at that moment (Heo et al., 21 May 2026). The operational pipeline is structured as Who $u$1 When $u$2 What: PersonaGate suppresses recommendations for players whose strategic consistency is empirically associated with superior outcomes, TimingGate uses the sign of $u$3 to decide when to switch, and ScoreFusion ranks candidate strategies by combining an adoptability signal with predicted transition quality ($u$4) (Heo et al., 21 May 2026).

Policy evaluation in this setting uses SwitchGap rather than the player’s observed action as optimal ground truth: $u$5 This measures how much better a policy’s approved switches turn out to be, compared to its rejected switches, among games where players did switch. The full pipeline achieves a SwitchGap of $u$6 percentage points at a recommendation rate of $u$7, and loss-triggered switchers benefit the most from subtype-conditioned guidance (Heo et al., 21 May 2026). One common misconception addressed here is that frequent historical switching furnishes an optimal reference policy; the framework explicitly rejects that premise because the most frequent switchers record the lowest win rates.

5. Exact-state attainment and benchmark win rates

In benchmarked interaction settings, a state-conditioned empirical win rate can be defined directly as the proportion of instances for which an agent reaches the exact target state. FineState-Bench formalizes this with four stage-wise metrics built from indicator functions: $u$8 and

$u$9

The resulting metrics are as follows (Ji et al., 30 Apr 2026).

Metric	Definition
SR@Loc	$b$0
SR@Int	$b$1
ES-SR@Loc	$b$2
ES-SR@Int	$b$3

Because the benchmark focuses on exact-state attainment after a single point-action, the principal summary metric is

$b$4

If the evaluation is aggregated across platforms, the global value is the instance-weighted average

$b$5

The benchmark’s construction is designed to make this exact-state win rate meaningful rather than approximate. FineState-Bench contains $b$6 static, single-step instances: Desktop $b$7, Web $b$8, Mobile $b$9. It spans four interaction families and $\mathrm{stay\_baseline}(s,u,b)$0 UI component types, and every instance is manually verified to include an unambiguous natural-language instruction pair, an exact, machine-readable $\mathrm{stay\_baseline}(s,u,b)$1, and dual bounding boxes $\mathrm{stay\_baseline}(s,u,b)$2 under current and target configurations so that layout changes during the operation do not invalidate evaluation (Ji et al., 30 Apr 2026). By construction, each $\mathrm{stay\_baseline}(s,u,b)$3 is a precise state label, enabling exact match verification of $\mathrm{stay\_baseline}(s,u,b)$4.

The benchmark also illustrates that a state-conditioned empirical win rate can be diagnostic rather than merely summative. Exact goal-state success remains low: ES-SR@Int peaks at $\mathrm{stay\_baseline}(s,u,b)$5 on Web and $\mathrm{stay\_baseline}(s,u,b)$6 on average across platforms. The Visual Diagnostic Assistant optionally augments the agent’s input with a short Description and a tight Localization Hint bounding box $\mathrm{stay\_baseline}(s,u,b)$7, and controlled comparisons then measure the recoverable gain

$\mathrm{stay\_baseline}(s,u,b)$8

With VDA localization hints, Gemini-2.5-Flash gains $\mathrm{stay\_baseline}(s,u,b)$9 ES-SR@Int points; the detailed summary reports a baseline ES-SR@Int of approximately $(m,n)$00 average over Mobile/Web/Desktop and approximately $(m,n)$01 with VDA, with platform-wise changes of $(m,n)$02 on Mobile, $(m,n)$03 on Web, and $(m,n)$04 on Desktop (Ji et al., 30 Apr 2026). This suggests that a low “win rate” may reflect visual grounding failure, exact control failure, or both, which is precisely why the benchmark decomposes success into stage-wise components rather than relying on final-task success alone.

6. Bayesian smoothing, calibration, and sparse-state regimes

When the state space is sparse, raw empirical win rates can be unstable. The soccer model of Robberechts et al. addresses this by learning a Bayesian posterior predictive over future goals and mapping it to win, tie, and loss probabilities. For each team, future goals after frame $(m,n)$05 are modeled as

$(m,n)$06

with scoring intensities

$(m,n)$07

and weakly informative normal priors plus a random walk on $(m,n)$08 (Robberechts et al., 2019). The home-win probability is then

$(m,n)$09

with tie and loss defined analogously.

The technical summary explicitly contrasts this with a naive discretized estimator

$(m,n)$10

and notes that because soccer is low-scoring and very few games occupy exactly the same state, $(m,n)$11 is extremely noisy. The Bayesian formulation can therefore be viewed as a hierarchical smoothing of sparse cell counts. If one wished to smooth a binary raw-count win rate directly, the supplied summary gives the Beta-smoothed posterior mean

$(m,n)$12

with the multinomial extension obtained by replacing Beta with a Dirichlet prior (Robberechts et al., 2019).

Inference is performed with PyMC3’s automatic differentiation variational inference rather than exact MCMC: $(m,n)$13 gradient-descent steps, minibatches of $(m,n)$14 games, and $(m,n)$15 posterior samples for posterior predictives, requiring approximately $(m,n)$16 minutes on a 4-core Xeon with $(m,n)$17 GB RAM for approximately $(m,n)$18 games (Robberechts et al., 2019). Evaluation uses reliability diagrams, multi-class expected calibration error, and Ranked Probability Score. The Bayesian model reports ECE values of $(m,n)$19 in the first half, $(m,n)$20 in the second half, $(m,n)$21 in the final $(m,n)$22 of the match, and $(m,n)$23 overall, outperforming the listed logistic regression, multi-LR, and random forest baselines in overall calibration and remaining well-calibrated even near full time (Robberechts et al., 2019).

Across the supplied sources, three interpretive cautions recur. First, sparse state occupancy makes unsmoothed empirical win rates unreliable, motivating Bayesian or hierarchical regularization in soccer (Robberechts et al., 2019). Second, observed historical choices should not automatically be treated as optimal ground truth, which is why Clash Royale introduces SwitchGap and emphasizes that the most frequent switchers record the lowest win rates (Heo et al., 21 May 2026). Third, final-task success alone may obscure the source of failure, which is why FineState-Bench separates localization, interaction, and exact-state attainment (Ji et al., 30 Apr 2026). Taken together, these formulations present state-conditioned empirical win rate not as a single statistic, but as a general methodology for assigning success probabilities or success frequencies to formally defined states, with exact, empirical, and Bayesian variants chosen according to the structure and sparsity of the problem.