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Game theory is a fascinating field of mathematics and economics that studies strategic situations. These involve multiple decision-makers, such as players in a game. Each player tries to maximize their own payoff. In the context of "Rock, Paper, Scissors" modified for three players, each player selects an option based on their strategy to win.

The game becomes a strategic battle, as each player's choice affects not only their own outcome but also the outcome of others. The aim is to understand how players make decisions and to predict which strategies will emerge as dominant.

The players do not know in advance what the others will pick. This makes their choice a part of strategic decision-making. This situation is a classic example of a non-cooperative game where the focus is on competition.

Expected value is a key concept in probability and statistics. It quantifies the average outcome of a random event after many repetitions. In the modified Rock, Paper, Scissors game, each player's win condition has a certain probability.

For a game to be considered fair, the expected value of winning for each player should be the same. The expected value is calculated by multiplying the probability of each outcome by the payoff for that outcome. Then, sum these values up.

For example, if Player B has a 4/9 probability of winning $1, then B's expected value is (4/9) × $1 = $4/9. Similarly, Player C’s expected value with a 2/9 probability and a $3 payout is (2/9) × $3 = $6/9. Making the game fair involves setting Player A's payout such that the expected value is equal to $4/9 or $6/9.

A fair game is one where each player has the same expected outcome over time, ensuring no player has a systematic advantage.

In the given problem, fairness requires that the expected winnings of Player A, Player B, and Player C are identical. This is because each player's chance of winning varies and is represented by different payoffs.

To determine a fair payout for Player A, we set their expected value of winning equal to those of players B and C. This involves solving the equation of expected values derived from their respective probabilities and payouts. If Player B receives $1 and Player C receives $3, this needs to balance Player A’s expected value, resulting in Player A needing a payout of $4 for fairness.

Maintaining fairness is crucial for games to be engaging and enjoyable, preventing any player from feeling disadvantaged over time.

Combinatorics is an area of mathematics focused on counting, arrangement, and combination of sets. It plays a significant role in determining the number of possible outcomes and the probability associated with each one in games.

For the modified Rock, Paper, Scissors, combinatorics helps us calculate the different scenarios in which each player wins. By counting possibilities, we see Player A can win if all choose the same object: 3 outcomes (e.g., all Rock, all Paper, all Scissors).

Player B's win condition is more complex, involving exactly two players choosing the same and one choosing differently. This accounts for 12 combinations, calculated using permutations.

Player C wins when all three players pick different items, which occurs in 6 ways, thanks to the factorial counting rule. Understanding these outcomes and their likelihoods requires combinatorial knowledge. This makes it crucial in predicting probabilities in strategic games.