Question

Give an example of a game that is strictly competitive with respect to pure strategies but is not strictly competitive with respect to mixed strategies.

Short answer

Rock-Paper-Scissors is competitive with pure, but not with mixed strategies.

Step-by-step solution

Understanding Strictly Competitive Games

A strictly competitive game is one where the gain of one player equates to a loss for the other. In pure strategies, this is clear cut, as each player has a specific move that directly counters the opponent.

Identifying Pure Strategy Competitiveness

Consider a simple zero-sum game like Rock-Paper-Scissors. In pure strategies, choosing Rock beats Scissors, resulting in a clear win/loss situation, making it strictly competitive in this form.

Exploring Mixed Strategies

Mixed strategies involve players choosing probabilistic combinations of strategies. In Rock-Paper-Scissors, if each player chooses randomly among Rock, Paper, and Scissors, the expected outcome is a tie, not a strict win/loss relationship.

Conclusion with an Example

Rock-Paper-Scissors is strictly competitive with pure strategies as each choice beats another and loses to a different one. However, when players adopt mixed strategies with equal probabilities, the competitiveness vanishes as the expected value is neutral.

Key concepts

Strictly Competitive Games

In game theory, a strictly competitive game occurs when a player's gain is always precisely matched by the opponent's loss. This type of game often leads to intense competition where one player's benefit directly translates to the other player's detriment.
In simple terms, if one player is winning, the other must be losing. Each action a player takes aims to maximize their gain while simultaneously minimizing the opponent's advantage.
A classic example is a zero-sum game like chess, where one player's victory means the other's defeat. The strictly competitive nature stems from the necessity for one player's success to come at the complete cost of the other.

Pure Strategies

Pure strategies in game theory involve making definitive, consistent choices. These are straightforward actions selected without any random deviation.
For instance, in a game like Rock-Paper-Scissors, choosing "Rock" every single round is a pure strategy. The outcome of pure strategies in strictly competitive games is often direct, resulting in clear winners and losers.
Players fully commit to one outcome without variability. This creates a predictable pattern, unlike mixed strategies which introduce randomness.

Mixed Strategies

Mixed strategies are a fascinating component of game theory. They involve players choosing among possible strategies according to a particular probability distribution. This means decisions aren't set in stone; instead, players switch between various strategies, keeping opponents guessing.
In the case of the game Rock-Paper-Scissors, employing a mixed strategy would mean each player opts for Rock, Paper, or Scissors with equal probability. This unpredictability levels the playing field, often leading to a neutral expected outcome. Unlike pure strategies, the inclusion of randomness in mixed strategies blurs the lines of competition, offering no guaranteed winner.
This feature of mixed strategies enriches game theory by demonstrating strategic diversity and the balance of power when certainty is removed from decision-making.

Zero-Sum Game

A zero-sum game is a pivotal concept in game theory, where the gains and losses are balanced perfectly among the players involved. In this scenario, one player's advantages directly subtract from the other's, making the net change in value zero.
Consider a game of poker: if one player wins money, the other loses the equivalent amount. This balance encapsulates the essence of zero-sum games.
These games are a type of strictly competitive game, where the stakes are such that the total payoff is a constant value, and any gain for one party is a loss for the other. Understanding zero-sum games is crucial as they often underpin strategic thinking and competitive dynamics in various real-world situations, such as economics, sports, and conflict resolution.