# Rock-Paper-Scissors

## <mark style="color:purple;">Application in Luck measurement</mark>

To measure a person's Luck Index through Rock, Paper, Scissors (RPS), we recognize it as a chance-based game but account for psychological influences that introduce non-randomness. Assuming that over many games, the impact of skill or strategy diminishes, allowing outcomes to more accurately represent chance.

### <mark style="color:purple;">Large Number of Trials</mark>

Measuring the Luck Index via RPS game requires numerous trials, adhering to the Law of Large Numbers, which states that outcomes will align with their expected probabilities over many trials. Given RPS's equal likelihood for win, lose, or draw in a random scenario, each has a theoretical 1/3 probability, assuming true randomness.

### <mark style="color:purple;">Quantifying Outcomes</mark>

For each participant, we record outcomes across these trials, classifying each as a win, loss, or draw. The Luck Index, in this context, would focus on the proportion of wins, adjusted for the expected win rate in a fair game.

### <mark style="color:purple;">Expected vs. Actual Outcomes</mark>

The expected win rate in a truly random game of RPS is $$1/31/3$$. However, actual outcomes may deviate from this expectation. By comparing the actual win rate to the expected win rate, we can begin to quantify an individual's luck:

* **Actual Win Rate (AWR)**: The number of wins divided by the total number of games played (excluding draws).
* **Expected Win Rate (EWR)** for a perfectly balanced game: $$1/3$$.

#### 4. Calculating the Luck Index

To adapt our [Luck Index](/luckaton-and-science/luck-index.md) formula to the context of RPS outcomes:

$$LI= \frac{AWR-EWR}{1-EWR}$$

This formula adjusts each person's win rate by the expected win rate, allowing us to quantify deviations from what would be anticipated by chance alone.

### <mark style="color:purple;">Statistical Significance</mark>

To ascertain that the observed outcomes (and by extension, the calculated Luck Index) are not merely the result of random fluctuations, Chi-square statistical test is being used to evaluate the significance of the deviation of observed outcomes from expected outcomes.

### <mark style="color:purple;">Practical Considerations</mark>

* **Sample Size**: A sufficiently large number of trials is essential to ensure reliability and to mitigate the influence of short-term fluctuations.
* **Player Behavior**: Since human players may not choose outcomes entirely at random, Luckaton prompts to randomize player strategies to minimize the influence of skill and is limiting the number of times 2 players can play together within a set time frame (e.g. 24 hours).
* **Fairness**: Luckaton ensures that the game conditions are identical for all participants to prevent external factors from influencing the outcomes.

By following this approach, it becomes feasible to scientifically justify the use of the Luck Index calculated from RPS outcomes. It's important to remember, however, that while this method can offer insights into the luck factor in controlled conditions, luck is a complex and multifaceted concept that may not be fully captured by any single metric or game outcome.

***

### <mark style="color:purple;">One of the most ancient and well-known games</mark>

Rock-Paper-Scissors, or shoushiling, dates back to China's Han dynasty (206 BC – 220 AD), first mentioned in Xie Zhaozhi Ming dynasty work 'Wuzazu' and also referenced in Li Rihua's 'Note of Liu Yan Zhai.'


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