State-Dependent Asymmetry in Soft-Pity Gacha Waiting-Time Models: Exact Recurrences, Tail Risk, and Featured-Target Extensions

1. Introduction

Randomized reward mechanisms have become a prominent feature of contemporary digital games. In an idealized constant-rate model, attempts are exchangeable in the following limited sense: conditional on no earlier rare item, the next draw has the same success probability as any previous draw. Pity mechanisms remove this draw-count homogeneity. A hard pity rule truncates the tail by forcing success at a fixed counter value, whereas a soft-pity rule raises the success probability before the hard guarantee. Featured-target rules introduce a second state variable, since the probability of receiving the desired target may depend on whether a guarantee is active.

This paper uses symmetry and asymmetry in this probabilistic sense. The point is not to develop a group-theoretic or physical theory of symmetry breaking, but to use the constant-hazard model as a reference case and to quantify how state dependence changes the induced waiting-time distribution. The resulting asymmetry is visible in the hazard function, the probability mass function, and upper-tail risk measures. This viewpoint is useful because the mean waiting time alone conceals much of the risk structure: two mechanisms with similar expectations may have different quantiles, expected excess values, and completion probabilities.

The broader psychological and consumer-protection literature provides the motivation for studying randomized reward mechanisms. Loot boxes and related random-item purchases have attracted attention because they share structural features with gambling-like reinforcement schedules, and because spending on such mechanisms has been associated with problem-gambling indicators in some samples [1,2,3,4]. The empirical evidence is not identical across contexts. For instance, Xiao, Fraser, and Newall report weak links between gambling and loot-box expenditure in a preregistered Chinese sample, while also discussing probability disclosures and pity timers [5]. More recent work has examined compliance with probability-disclosure rules in South Korea [6]. Outside video games, blind-box consumption has also been studied through curiosity, impulse buying, and price-consciousness mechanisms [7].

The present paper addresses a narrower mathematical question. Given a pity schedule, how can one compute the mean, variance, distribution, quantiles, tail probabilities, and distributional asymmetry of the waiting time for rare outcomes? The analysis is therefore model based. Its numerical values come from the stated transition rule rather than from observed draw histories, and simulation is used only to check the implementation of the exact recurrences.

Such a format is standard in applied probability. Finite Markov chains, absorption times, and first-step recurrences are routinely used to turn a specified stochastic mechanism into computable expectations and distributions [13,14,15,16]. Related work on hitting-time distributions for finite or skip-free Markov chains likewise emphasizes explicit recurrences, transforms, and numerical algorithms [9,10,11].

The model considered here has one primary state variable: the pity counter, i.e., the number of consecutive failures since the last rare item. A success resets this counter; a failure increases it by one; and a hard pity limit bounds the waiting time. A soft-pity region is represented by increasing the success probability before the hard guarantee. First-step analysis gives backward recurrences for the expected absorption time. The law of total variance gives a companion recurrence for the variance. A probability-mass recursion then yields the exact waiting-time distribution and, by convolution, exact distributions for sums of independent stages.

The contribution is computational and probabilistic. The paper gives a compact derivation for a soft-pity waiting-time model, interprets pity and guarantee rules as state-dependent departures from a constant-rate baseline, implements exact distributional computation, and reports risk summaries that are not visible from the mean alone. Commercial gacha systems may include additional mechanisms such as featured-item probability-up, 50/50 guarantees, duplicate outcomes, weapon-banner epitomized-path rules, cross-banner carry-over, and player stopping rules. Related economic work has considered gacha-like monetization through prospect theory and optimal pricing [8]. Those broader questions require additional state variables and behavioral assumptions; here they appear only as extensions or illustrative numerical variants.

The paper is organized as follows. Section 2 defines the pity-counter model and the representative transition schedule, including its symmetry/asymmetry interpretation. Section 3 derives exact recurrences for expectation, variance, and the probability mass function. Section 4 reports the core numerical results. Section 5 adds scaling experiments, upper-tail risk metrics, distributional asymmetry diagnostics, approximation checks, and a featured-target numerical extension. Section 6 records further modeling extensions. Section 7 discusses limitations and reproducibility.

2. Model

2.1. Pity-Counter State

Let the pity counter be

$$t\in \{0,1,\dots ,89\},$$

where t is the number of consecutive failures since the most recent rare item. A draw from state t succeeds with probability $p_t$. On success, the current stage ends and the pity counter would reset to zero for the next stage. On failure, the counter increases from t to $t+1$. At $t=89$, the next draw is a hard guarantee and therefore succeeds with probability one.

The representative schedule studied in this paper is

$$p_t=\left\{\begin{array}{cc}0.006,\hfill & 0\le t<73,\hfill \\ 0.006+(t-72)\times 0.0585,\hfill & 73\le t<89,\hfill \\ 1,\hfill & t=89.\hfill \end{array}\right.$$

(1)

The base rate 0.006 is consistent with official disclosures for five-star drops in a well-known gacha title [12]. The soft-pity threshold and linear slope in (1), however, should not be interpreted as official parameters. They are used as a representative working schedule; if a different schedule is supplied, the same recurrences apply without structural change.

Figure 1. Representative soft-pity success probability $p_t$ as a function of the pity counter. The probability is flat before the soft-pity region, rises sharply inside the ramp, and reaches the hard guarantee at $t=89$.

Figure 1. Representative soft-pity success probability $p_t$ as a function of the pity counter. The probability is flat before the soft-pity region, rises sharply inside the ramp, and reaches the hard guarantee at $t=89$.

2.2. Symmetry and Asymmetry of the Transition Rule

The constant-rate no-pity model has a homogeneous transition structure: before absorption, the success probability is invariant with respect to the number of previous failures. In this limited but useful sense, every non-absorbed draw is probabilistically symmetric with every other non-absorbed draw. This is the only notion of symmetry used in the present paper. Hard pity and soft pity break this draw-count homogeneity. Hard pity preserves a constant hazard until the terminal counter and then imposes a boundary asymmetry at the guarantee. Soft pity breaks the symmetry earlier by assigning different transition probabilities to different pity-counter states.

Figure 2 compares the hard-pity-only hazard with the representative soft-pity hazard. The mathematical consequence of this state dependence is not merely a smaller expected value: it changes the shape, skewness, entropy, and upper-tail behavior of the absorption-time distribution. These effects are quantified later in Table 1 and Figure 3.

2.3. Single-Stage and Multi-Stage Quantities

A stage begins at $t=0$ and ends at the first success. Let $\tau$ denote the number of draws in one stage. Because success resets the pity counter, repeated attempts to obtain independent rare items can be modeled as independent copies of $\tau$, provided that no additional mechanism changes the stage distribution. For m independent rare items, define

$$T_m={\tau }_1+{\tau }_2+\dots +{\tau }_m,$$

(2)

where ${\tau }_1,\dots ,{\tau }_m$ are i.i.d. copies of $\tau$. The numerical section focuses on $m=10$ to illustrate distributional convolution, but the formulas hold for arbitrary fixed m.

Remark 1

(Scope of independence). The independence in (2) is not a claim about all commercial gacha systems. It is valid only for the simplified model in which each rare-item acquisition resets the counter and the next stage has the same transition schedule. If featured-item guarantees or duplicate collection are added, the stage distribution changes with the collection state, and the enlarged process is still Markovian but no longer a sum of identical independent stages.

3. Exact Recurrences

3.1. Expectation

Let $E_t$ be the expected number of additional draws required to obtain a success when the current pity counter is t.

Proposition 1

(Expectation recurrence). The expected hitting times satisfy

$$E_{89}=1$$

(3)

and, for $t=0,1,\dots ,88$,

$$E_t=1+(1-p_t)E_{t+1}.$$

(4)

Proof. 

From state t, one draw is made. With probability $p_t$, the draw succeeds and no further draws are needed. With probability $1-p_t$, the draw fails and the process continues from state $t+1$. Therefore

$$E_t=1+p_t·0+(1-p_t)E_{t+1}=1+(1-p_t)E_{t+1}.$$

At $t=89$, the next draw succeeds with probability one, so $E_{89}=1$. □

The recurrence can also be written explicitly as a finite survival sum. If $S_j=P(\tau >j)$, then

$$E_0={\displaystyle \sum _{j=0}^{89}}P(\tau >j)={\displaystyle \sum _{j=0}^{89}}{\displaystyle \prod _{r=0}^{j-1}}(1-p_r),$$

(5)

with the empty product interpreted as one. The backward recurrence is usually more convenient computationally, but (5) highlights that the hitting time has bounded support.

3.2. Variance

Let $V_t$ be the variance of the number of additional draws needed from state t.

Proposition 2

(Variance recurrence). The variances satisfy

$$V_{89}=0$$

(6)

and, for $t=0,1,\dots ,88$,

$$V_t=(1-p_t)V_{t+1}+p_t(1-p_t)E_{t+1}^2.$$

(7)

Proof. 

Let $X_t$ be the additional number of draws to success from state t. Write

$$X_t=1+Z_t,$$

where $Z_t=0$ if the first draw from state t succeeds, and $Z_t=X_{t+1}$ if the first draw fails. Conditioning on the first draw outcome and using the law of total variance gives

$$\begin{array}{cc}\hfill Var\left(X_t\right)& =Var\left(Z_t\right)\hfill \\ \hfill & =\mathbb{E}\{Var(Z_t\mid \mathrm{first}\mathrm{outcome})\}+Var\left\{\mathbb{E}(Z_t\mid \mathrm{first}\mathrm{outcome})\right\}.\hfill \end{array}$$

The first term equals $(1-p_t)V_{t+1}$, since the successful branch has zero conditional variance and the failed branch has variance $V_{t+1}$. The conditional expectation of $Z_t$ equals 0 on the successful branch and $E_{t+1}$ on the failed branch. Hence its variance is

$$p_t(1-p_t)E_{t+1}^2.$$

Adding the two terms proves (7). The boundary condition follows because the hitting time from $t=89$ is deterministically equal to one. □

For m independent stages,

$$\mathbb{E}\left[T_m\right]=m{E}_0,Var\left(T_m\right)=m{V}_0.$$

(8)

3.3. Probability Mass Function

Expectation and variance summarize only part of the risk profile. Quantiles and tail probabilities require the full distribution. Let

$$f_t\left(j\right)=P(\tau =j\mid \mathrm{current}\mathrm{pity}\mathrm{counter}t).$$

Proposition 3

(PMF recursion). The probability mass function satisfies

$$\begin{array}{cc}\hfill f_{89}\left(1\right)& =1,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill f_t\left(1\right)& =p_t,0\le t<89,\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill f_t\left(j\right)& =(1-p_t)f_{t+1}(j-1),j\ge 2,0\le t<89.\hfill \end{array}$$

(11)

Proof. 

If $t=89$, the hitting time is exactly one draw. If $t<89$, a hitting time equal to one occurs exactly when the first draw succeeds. For $j\ge 2$, the first draw must fail and the remaining hitting time from state $t+1$ must be $j-1$. This gives (). □

Once $f_0$ is obtained, the distribution of $T_m$ is the m-fold discrete convolution

$$P(T_m=n)=\left(f_0^{*m}\right)\left(n\right).$$

(12)

Because $\tau \le 90$, the support of $T_m$ is contained in $\{m,m+1,\dots ,90m\}$. Direct convolution is therefore exact and computationally trivial for the values of m considered here.

3.4. Reproducibility Algorithm

The following pseudocode is sufficient to reproduce the numerical results in the paper.

Step 1.

Define $p_t$ by (1) for $t=0,\dots ,89$.

Step 2.

Initialize $E_{89}=1$ and compute $E_{88},E_{87},\dots ,E_0$ by (4).

Step 3.

Initialize $V_{89}=0$ and compute $V_{88},V_{87},\dots ,V_0$ by (7), using the already computed values of $E_t$.

Step 4.

Initialize $f_{89}\left(1\right)=1$, then compute $f_t\left(j\right)$ for decreasing t using ()–().

Step 5.

Repeatedly convolve $f_0$ with itself to obtain the PMF of $T_m$.

Step 6.

Compute tail probabilities and quantiles directly from the cumulative distribution function.

These steps are deterministic once the transition schedule has been specified.

4. Numerical Results

This section evaluates the recurrences in Section 3 under Equation (1). The values are distributional consequences of that schedule.

4.1. Single-Stage Moments

Solving (4) and (7) under (1) gives

$$E_0\approx 62.3377,V_0\approx 592.4345,\sqrt{V_0}\approx 24.3400.$$

(13)

Thus one rare item requires about 62.34 draws on average under the representative soft-pity schedule.

It is useful to compare this value with two simple baselines. If there were no pity mechanism at all and every draw had success probability $0.006$, then the waiting time would be geometric with mean

$${\displaystyle \frac{1}{0.006}}\approx 166.67.$$

(14)

If there were a hard guarantee at 90 draws but no soft-pity ramp, then $p_t=0.006$ for $t=0,\dots ,88$ and $p_{89}=1$, which gives an expected waiting time of approximately 69.70 draws. The representative soft-pity ramp therefore reduces the mean by about 62.6% relative to the uncapped geometric baseline and by about 10.6% relative to the hard-pity-only baseline.

Table 2. Single-stage benchmark comparison under base rate 0.006.

Model	Expected draws	Comment
Uncapped no-pity geometric model	166.67	No guarantee
Hard pity only at 90 draws	69.70	No soft-pity ramp
Representative soft-pity model (1)	62.34	Linear ramp before hard pity

Table 2. Single-stage benchmark comparison under base rate 0.006.

Model	Expected draws	Comment
Uncapped no-pity geometric model	166.67	No guarantee
Hard pity only at 90 draws	69.70	No soft-pity ramp
Representative soft-pity model (1)	62.34	Linear ramp before hard pity

Figure 4. Expected waiting time under three baselines: uncapped geometric draws, hard pity without a soft-pity ramp, and the representative soft-pity schedule.

Figure 4. Expected waiting time under three baselines: uncapped geometric draws, hard pity without a soft-pity ramp, and the representative soft-pity schedule.

4.2. Expected Remaining Draws Inside the Soft-Pity Region

Table 3 reports selected values of $E_t$. The reduction column compares $E_t$ with $E_0$.

The table shows why the phrase “close to pity” has a precise mathematical meaning in this model. By counter value 72, the expected remaining waiting time has fallen below six draws; by counter value 80, it is below two draws. These values are not simulations; they are direct outputs of the exact backward recurrence. Figure 5 and Figure 6 visualize the same recurrence outputs over the full counter range.

4.3. Single-Stage Distributional Shape

The exact PMF recursion gives more information than the first two moments. Figure 7 shows the single-stage distribution of $\tau$, while Figure 8 gives the corresponding cumulative and survival functions. These plots make clear that the model has two distinct features: an initially long low-probability region and a sharp accumulation of mass around the soft-pity and hard-pity region.

4.4. Ten-Item Distribution

For $m=10$, (8) gives

$$\mathbb{E}\left[T_{10}\right]\approx 623.3766,Var\left(T_{10}\right)\approx 5924.3447,sd\left(T_{10}\right)\approx 76.9698.$$

(15)

The exact convolution distribution gives the quantiles in Table 4.

Figure 9. Exact distribution of $T_{10}$, obtained by tenfold convolution of the single-stage PMF. The distribution is bounded between 10 and 900 draws.

Figure 9. Exact distribution of $T_{10}$, obtained by tenfold convolution of the single-stage PMF. The distribution is bounded between 10 and 900 draws.

Figure 10. Exact quantile curve of $T_{10}$. The upper tail illustrates why quantiles provide information that is not visible from the mean alone.

Figure 10. Exact quantile curve of $T_{10}$. The upper tail illustrates why quantiles provide information that is not visible from the mean alone.

Figure 11. Exact right-tail function $P(T_{10}\ge b)$ for ten independent rare items. This figure converts a draw budget b into a non-completion probability under the simplified model.

Figure 11. Exact right-tail function $P(T_{10}\ge b)$ for ten independent rare items. This figure converts a draw budget b into a non-completion probability under the simplified model.

Table 5. Exact right-tail probabilities for $T_{10}$.

Threshold b	$P(T_{10}\ge b)$
650	0.39268
700	0.16794
724	0.08755
750	0.03692
778	0.00730
800	0.00004

Table 5. Exact right-tail probabilities for $T_{10}$.

Threshold b	$P(T_{10}\ge b)$
650	0.39268
700	0.16794
724	0.08755
750	0.03692
778	0.00730
800	0.00004

For example, the exact right-tail probability at 724 draws is

$$P(T_{10}\ge 724)\approx 0.08755.$$

(16)

Since $724-623.38\approx 100.62$, this event is approximately 1.31 standard deviations above the mean, not a two-standard-deviation event. A one-sided two-standard-deviation threshold is approximately

$$623.38+2\left(76.97\right)\approx 777.32,$$

and the exact tail probability at 778 draws is approximately

$$P(T_{10}\ge 778)\approx 0.00730.$$

(17)

These values are obtained by dynamic-programming convolution. A Monte Carlo run with ${10}^6$ replications differs by less than 0.001 for (16), as shown in Figure 12.

4.5. Auxiliary Sensitivity Analysis

The representative schedule in Equation (1) is the main object of this paper. To illustrate how the same code can compare alternative designs, we also compute an auxiliary sensitivity experiment in which the start of a linear ramp is varied while the hard guarantee remains fixed at 90 draws. In this auxiliary experiment, the ramp is re-scaled so that it reaches probability one at the hard guarantee. The resulting values should not be interpreted as estimates of any particular commercial system; they only demonstrate how the recurrence framework can be used to compare schedules.

Table 6. Sensitivity of the single-stage expected waiting time to the beginning of a linear ramp reaching hard pity at draw 90. This table uses an auxiliary ramp definition and is not the main schedule in Eq. (1).

Ramp start counter	$E_0$	$sd\left(\tau \right)$
60	54.98	19.95
65	57.91	21.64
70	60.71	23.33
73	62.34	24.34
76	63.91	25.34
80	65.92	26.64

Table 6. Sensitivity of the single-stage expected waiting time to the beginning of a linear ramp reaching hard pity at draw 90. This table uses an auxiliary ramp definition and is not the main schedule in Eq. (1).

Ramp start counter	$E_0$	$sd\left(\tau \right)$
60	54.98	19.95
65	57.91	21.64
70	60.71	23.33
73	62.34	24.34
76	63.91	25.34
80	65.92	26.64

Figure 13. Auxiliary sensitivity analysis in which the start of a linear ramp is varied while the hard guarantee remains at 90 draws. This figure is not a calibration claim; it only illustrates how the recurrences can compare alternative schedules.

Figure 13. Auxiliary sensitivity analysis in which the start of a linear ramp is varied while the hard guarantee remains at 90 draws. This figure is not a calibration claim; it only illustrates how the recurrences can compare alternative schedules.

5. Extended Numerical Experiments

The preceding section reports the core quantities for the simplified rare-item model. This section expands the numerical analysis in four directions: scaling with the number of desired rare items, upper-tail risk metrics, approximation diagnostics, and an illustrative featured-target extension. All computations use the PMFs generated by the recurrences above or by explicitly stated auxiliary state spaces.

5.1. Scaling with the Number of Rare Items

For general m, the exact PMF of $T_m$ is obtained by repeated convolution. Table 7 and Figure 14 show how the mean and upper quantiles grow as m increases from 1 to 20. The mean grows linearly by construction, while upper quantiles include an additional dispersion term. A budget based only on $m{E}_0$ generally gives a completion probability well below high-confidence levels such as 90% or 95%.

Figure 15 plots $P(T_m\le b)$ over both the desired number of rare items and the available draw budget. This directly answers how likely a given budget is to finish a target count.

5.2. VaR, CVaR, and Expected Excess

Quantiles do not describe how large the draw count can be after the quantile is exceeded. Table 8 reports Value-at-Risk and a discrete upper-tail conditional mean for $T_{10}$. Here ${CVaR}_{\alpha }$ is used descriptively as $E[T_{10}\mid T_{10}\ge {VaR}_{\alpha }]$. Table 9 reports completion probability and expected positive excess $E\left[{(T_{10}-b)}_{+}\right]$.

Figure 16. Upper-tail risk metrics for $T_{10}$. The CVaR curve lies above the VaR curve because it averages over the tail beyond the quantile.

Figure 16. Upper-tail risk metrics for $T_{10}$. The CVaR curve lies above the VaR curve because it averages over the tail beyond the quantile.

5.3. Normal Approximation Diagnostics

Because $T_m$ is a sum of bounded independent variables, a normal approximation becomes plausible as m grows. However, exact convolution allows this approximation to be checked rather than assumed. For $T_{10}$, the maximum absolute CDF error of the continuity-corrected normal approximation is approximately 0.0280. Table 10 compares exact and normal-approximate tail probabilities, and Figure 17 shows the signed CDF error for several values of m.

5.4. Parameter Sensitivity

The representative schedule in Equation (1) is a modeling input. Figure 18 varies both the beginning of the soft-pity ramp and its slope. Figure 19 varies the base probability while holding the same general ramp form. They show how the distribution changes across alternative schedules.

5.5. Illustrative Featured-Target Extension

Finally, we include an illustrative extension that adds one commonly discussed target-item feature. Suppose that a rare item is the desired featured target with probability $q=0.5$ when the guarantee state is off. If the rare item is not featured, the pity counter resets and the next rare item is guaranteed to be featured. The Markov state is then $(t,g)$, where $g=0$ denotes no guarantee and $g=1$ denotes a featured guarantee. The target-acquisition state is absorbing for a single featured item. For $0\le t<89$, the transition probabilities from the non-guaranteed state are

$$\begin{array}{cccc}\hfill (t,0)& \to (t+1,0)\hfill & \hfill & \mathrm{with}\mathrm{probability}1-p_t,\hfill \\ \hfill (t,0)& \to \mathrm{target}\mathrm{absorption}\hfill & \hfill & \mathrm{with}\mathrm{probability}{p}_{t}q,\hfill \\ \hfill (t,0)& \to (0,1)\hfill & \hfill & \mathrm{with}\mathrm{probability}{p}_t(1-q).\hfill \end{array}$$

(18)

The first transition is an ordinary non-rare draw, the second is a rare draw that is featured, and the third is a rare draw that is not featured; in the third case the pity counter resets and the guarantee state is switched on. At $t=89$, the same formulas apply with $p_{89}=1$. From the guaranteed state,

$$\begin{array}{cccc}\hfill (t,1)& \to (t+1,1)\hfill & \hfill & \mathrm{with}\mathrm{probability}1-p_t,\hfill \\ \hfill (t,1)& \to \mathrm{target}\mathrm{absorption}\hfill & \hfill & \mathrm{with}\mathrm{probability}{p}_t.\hfill \end{array}$$

(19)

After a featured target has been obtained, the repeated-target calculations below assume that the next target begins again from $(0,0)$, i.e., with pity reset and no active featured guarantee. This reset convention is part of the auxiliary model and is stated explicitly because other conventions would lead to different multi-target distributions.

Under this illustrative 50/50 extension, starting without a guarantee gives an expected target-acquisition time of approximately 93.51 draws, with standard deviation approximately 43.13, a 90% quantile of 156, and a 95% quantile of 158. Table 11 reports the one-target summaries, and Table 12 gives repeated featured-target summaries under the convention that each featured success resets the guarantee state to no guarantee.

Figure 20. PMF comparison between obtaining any rare item and obtaining a featured target from the no-guarantee state under the illustrative 50/50 extension.

Figure 20. PMF comparison between obtaining any rare item and obtaining a featured target from the no-guarantee state under the illustrative 50/50 extension.

Figure 21. CDF comparison between any-rare acquisition and featured-target acquisition. The featured-target distribution has support up to 180 draws because one may first lose the 50/50 and then require a guaranteed rare item.

Figure 21. CDF comparison between any-rare acquisition and featured-target acquisition. The featured-target distribution has support up to 180 draws because one may first lose the 50/50 and then require a guaranteed rare item.

Figure 22. Mean and upper quantiles for repeated featured-target acquisition under the illustrative extension.

Figure 22. Mean and upper quantiles for repeated featured-target acquisition under the illustrative extension.

5.6. Distributional Asymmetry Diagnostics

The symmetry-breaking interpretation can also be inspected directly at the level of the waiting-time distribution. Table 1 reports compact distributional summaries for three model variants. The hard-pity-only and soft-pity distributions are both bounded, but the soft-pity ramp shifts probability mass toward the ramp region and changes the entropy and skewness of the PMF. The featured-target extension introduces an additional binary state asymmetry: starting from the no-guarantee state, the target-acquisition time is a mixture of one rare-item cycle and two rare-item cycles. This mixture explains why its skewness differs sharply from the one-stage rare-item waiting-time distributions. The negative skewness of the hard-pity-only distribution is also not an error: with bounded support and probability mass accumulating near the hard upper limit, the longer left-side spread relative to the terminal mass produces a left-skewed PMF.

6. Possible Extensions

The simplified model is useful because it isolates a single mathematical mechanism. Nevertheless, many realistic systems require larger state descriptions. This section records how the present framework can be extended without pretending that those extensions have already been analyzed in the numerical results.

6.1. Featured-Item Guarantees

A common featured-item mechanism can be represented by adding a binary guarantee state $g\in \{0,1\}$. Here $g=0$ means that the next rare item has only a stated featured-item probability, while $g=1$ means that the next rare item is guaranteed to be featured. The Markov state then becomes $(t,g)$. A non-rare draw changes $(t,g)$ to $(t+1,g)$. A rare draw resets the pity counter to zero and changes the guarantee state according to whether the featured item was obtained.

This enlarged chain is still finite and can be handled by the same first-step method. However, the single-stage distribution is no longer identical to the one studied above unless the initial guarantee state is fixed and the target is defined carefully. Therefore the numerical values in Section 4 should not be interpreted as featured-item acquisition times.

6.2. Duplicate Collection

Suppose a player seeks to collect all n distinct rare items and already owns k of them. If each rare item is equally likely conditional on a rare draw, the state must include both k and t. The transition structure is not obtained by simply replacing $p_t$ with $p_t(n-k)/n$, because a duplicate rare item resets the pity counter even though the collection count does not increase. A correct schematic transition is

$$\begin{array}{cccc}\hfill (k,t)& \to (k+1,0)\hfill & \hfill & \mathrm{with}\mathrm{probability}{p}_t{\displaystyle \frac{n-k}{n}},\hfill \\ \hfill (k,t)& \to (k,0)\hfill & \hfill & \mathrm{with}\mathrm{probability}{p}_t{\displaystyle \frac{k}{n}},\hfill \\ \hfill (k,t)& \to (k,t+1)\hfill & \hfill & \mathrm{with}\mathrm{probability}1-p_t.\hfill \end{array}$$

This observation is important because treating duplicate rare items as ordinary failures would systematically distort the expected collection time.

6.3. Budget and Risk Metrics

Once the exact distribution of $T_m$ is known, one can compute budget-risk quantities without additional modeling assumptions. For a budget of b draws, the non-completion probability is simply

$$P(T_m>b).$$

Similarly, Value-at-Risk can be read from the quantile table, and Conditional Value-at-Risk can be computed from the exact tail distribution. These quantities may be more informative than the mean alone because the player’s experience is governed by a realized tail event rather than by the expectation.

7. Discussion and Limitations

The analysis has three main implications for state-dependent asymmetry in randomized reward models. First, replacing a constant hazard by a counter-dependent hazard changes the full waiting-time distribution, not merely its expectation. Exact convolution makes quantiles, upper-tail probabilities, and expected excess values available without simulation error. Second, a hard guarantee bounds the support but does not remove practical tail risk when several stages are accumulated. Third, modest changes in the soft-pity threshold or ramp slope can move upper quantiles, so numerical conclusions should be tied to the stated schedule.

The word asymmetry is used here in the finite-state probabilistic sense: the transition law depends on the draw count and, in the featured-target extension, on a binary guarantee state. This interpretation is intentionally narrower than group-theoretic or physical symmetry breaking. Its role is to organize the comparison between a homogeneous repeated-trial baseline and state-dependent reward rules, and to motivate distributional summaries that are useful for risk assessment.

The transition schedule in Equation (1) is a representative mathematical input. Official disclosures can support a base probability such as 0.006 for certain five-star drops [12], but the soft-pity ramp used here is illustrative. The main model excludes behavioral responses, heterogeneous players, promotional timing, currency conversion, banner-specific rules, and budget-dependent stopping. The featured-target calculation in Section 5 is an auxiliary state-space extension rather than a complete model of all probability-up or weapon-banner systems. Empirical questions about spending, harm, or regulatory effectiveness would require independently documented data and a separate statistical design.

8. Conclusion

This paper derived exact recurrences for a simplified soft-pity gacha waiting-time model and computed the corresponding waiting-time distributions. Under the representative schedule in Equation (1), the expected waiting time for one rare item is approximately 62.34 draws, compared with approximately 166.67 draws in an uncapped geometric model and 69.70 draws in a hard-pity-only model. For ten independent rare items, the expected total is approximately 623.38 draws, the standard deviation is approximately 76.97 draws, and the 90% quantile is 719 draws.

The main methodological point is that state-dependent reward rules are best summarized distributionally. Once the transition probabilities are specified, finite recursion and convolution give quantiles, right-tail probabilities, expected excess, target-count scaling, and asymmetry indicators directly. Future mathematical work can extend the state space to incorporate featured-item guarantees, duplicate collection, carry-over across banners, and budget-dependent stopping rules. Empirical calibration would require independently documented draw histories and is outside the scope of the present model calculation.