When Should We Mix It Up? How to win at Rock-Paper-Scissors and Penalty Shootouts!

When Should We Mix It Up? Exploring Mixed Strategy Methods in Game Theory

A version of this article appears in my book, Twisted Logic: Puzzles, Paradoxes, and Big Questions (Chapman and Hall/CRC Press, 2024).

INTRODUCTION

In game theory, the concept of mixed strategy arises when players face a decision-making situation where they do not have a dominant strategy. A dominant strategy is a strategy that is always better than any other strategy, regardless of the opponent’s choice. However, in some cases, players employ a mixed strategy, which involves randomising their choices to maximise their expected payoffs.

CREATING UNCERTAINTY

The purpose of employing a mixed strategy is to create a balanced approach that maximises expected payoffs. By introducing randomness into their decision-making, players can create uncertainty for their opponents and avoid predictability. This uncertainty makes it difficult for opponents to exploit any patterns or weaknesses in the player’s choices.

In situations where no dominant strategy exists, players use mixed strategies to find a Nash Equilibrium, which is a state where no player can improve their payoff by unilaterally deviating from their current strategy. Nash Equilibria often involve players randomising their choices, as this creates a balance where no player can gain an advantage by deviating from their strategy.

Mixed strategies offer a powerful tool for players to navigate complex strategic interactions. By incorporating randomness, players can mitigate the risk of being exploited by opponents who attempt to exploit predictable behaviour. Instead, mixed strategies introduce a level of unpredictability, making it challenging for opponents to determine the player’s intentions and respond optimally.

ROCK-PAPER-SCISSORS

To illustrate this, consider a simple example of a two-player game, such as Rock-Paper-Scissors. In this game, each player has three pure strategies: Rock, Paper, and Scissors. If one player randomises their choices by assigning equal probabilities to each strategy, they introduce uncertainty into the game.

For instance, Player A might choose to play Rock, Paper, or Scissors with equal probabilities of 1/3 each. In response, Player B might also choose to play Rock, Paper, or Scissors with equal probabilities of 1/3 each.

In a real-life high stakes environment, the strategy has in fact been different, at least in one high profile case. I refer to the year 2005, when the president of Japanese electronics giant Maspro Denkoh Corporation faced a significant dilemma regarding the auction of the company’s prestigious art collection. Valued at around $20 million, the decision of whether Christie’s or Sotheby’s, both historic auction houses, should handle the auction was a challenging one. Unable to decide, he resorted to the game of rock, paper, scissors. This choice was seen as a fair way to resolve the impasse between the two firms.

Christie’s approach to the challenge was meticulous; the president of Christie’s in Japan researched the psychology behind the game and even consulted children who suggested avoiding ‘rock’ as the initial throw due to its predictability. Their strategy was to start with ‘scissors’. This move relied on the idea that Sotheby’s would anticipate a ‘rock’ throw from Christie’s and thus choose ‘paper’ to counteract it.

If the game had been structured as a best-of-three, Christie’s could have adapted their strategy based on findings from the State Key Laboratory of Theoretical Physics in China, which suggest that winners do not randomise but in practice tend to stick with their winning choice in the subsequent round.

However, this single-round match left no room for redemption or strategic evolution. Representatives from both auction houses met at Maspro’s offices, where they wrote down their selections. Christie’s emerged victorious with ‘scissors’, defeating Sotheby’s ‘paper’. This led to Christie’s winning the right to auction off the Maspro collection.

The auction, aptly nicknamed “Scissors”, culminated in the sale of several important works, including one of Cézanne’s paintings, which alone sold for $11.7 million at Christie’s New York salesroom!

Back now to the world of imagination and consider a penalty awarded during a championship final.

THE PENALTY

In the 88th minute of the match, a penalty is awarded against the defending champions. The penalty taker in our simplified scenario has two options: aim straight or aim at a corner. Similarly, the goalkeeper has two choices: stand still or dive to a corner. The probabilities of scoring or saving the penalty are as follows.

PENALTY TAKER’S PROBABILITY OF SCORING

Aims Straight/Goalkeeper Stands Still: 30% chance of scoring.

Aims Straight/Goalkeeper Dives: 90% chance of scoring.

Aims at Corner/Goalkeeper Stands Still: 80% chance of scoring.

Aims at Corner/Goalkeeper Dives: 50% chance of scoring.

GOALKEEPER’S PROBABILITY OF SAVING

Stands still/Penalty taker aims straight: 70% chance of saving.

Stands still/Penalty taker aims at corner: 20% chance of saving.

Dives/Penalty taker aims straight: 10% chance of saving.

Dives/Penalty taker aims at corner: 50% chance of saving.

ABSENCE OF DOMINANT STRATEGIES

Neither the penalty taker nor the goalkeeper has a dominant strategy in this game. A dominant strategy would be a choice that is superior to any other strategy, regardless of the opponent’s choice. Since this is not the case, both players must consider the opponent’s strategy when deciding their own.

MIXED STRATEGY EQUILIBRIUM

Game theory suggests that in the absence of dominant strategies, players should adopt mixed strategies to maximise their expected payoffs. A mixed strategy involves randomising the choices according to specific probabilities.

For the penalty taker, the optimal mixed strategy in this scenario involves aiming for the corner with a two-thirds (2/3) probability and shooting straight with a one-third (1/3) probability. This ratio can be derived with a bit of algebra by finding the ratio where the chances of scoring are the same, regardless of the goalkeeper’s strategy.

Likewise, the goalkeeper’s optimal mixed strategy involves diving for the corner with a five-ninths (5/9) probability and standing still with a four-ninths (4/9) probability. This ratio ensures that the chance of saving the penalty is equal, regardless of the penalty taker’s choice.

IMPLEMENTATION CHALLENGES

To effectively employ a mixed strategy, it is essential to introduce an element of randomness into the decision-making process. In the context of a penalty shootout, this requires the ability to randomise choices effectively.

For example, the penalty taker could use a method such as noting the time on the match clock, having divided these up mentally into six sections. If it shows section 1, 2, 3, or 4, the penalty taker aims for the corner; if it’s section 5 or 6, they shoot straight. Anyway, you get the general idea. This approach ensures that the penalty taker maintains the desired probability (2/3 in this case) of aiming for the corner.

SUPPORTING STUDIES

Game theory suggests that goalkeepers should randomise in some way their dive direction to optimally counteract any choice by the penalty taker. This concept, a fundamental part of game theory, aligns with real-world findings published in the American Economic Review. Additionally, scholarly debates have explored the pros and cons of various shooting and diving strategies in soccer. For instance, a paper published in Psychological Science indicated a tendency for goalkeepers to dive more frequently to the right when their team was trailing. However, this pattern wasn’t observed when their team was leading or the score was tied. Non-random patterns have also been identified in tennis, including published evidence that even professional players tend to alternate serves too regularly, while the stage of the game was again something of a predictor.

These studies provide insights into how mixed strategies and deviations from randomness can impact outcomes and shed light on the behaviour of players in high-stakes decision-making situations.

COMPLEX SCENARIOS INVOLVING MULTIPLE PLAYERS

In complex scenarios involving multiple players, such as in corporate marketing, mixed strategies become even more crucial. Companies often employ mixed strategies in competitive pricing, product launches, or market entries to avoid predictability that competitors could exploit. For instance, a company might randomise the timing of its product launches or sales promotions to keep competitors off-balance. In a multi-player market, this unpredictability can be a significant advantage, as it complicates the decision-making process for competitors.

PRACTICAL APPLICATION OF MIXED STRATEGIES

Military Tactics: In military operations, mixed strategies can be employed to make it difficult for the enemy to anticipate and prepare.

Political Campaigns: Political strategists often use mixed strategies in campaign messaging and policy announcements to keep opponents and voters engaged and guessing.

Corporate Negotiations: Companies may use mixed strategies in negotiation tactics, alternating between hard-line and conciliatory approaches.

CONCLUSION: BACK TO THE GAME

In conclusion, game theory and the concept of mixed strategies offer valuable insights into decision-making scenarios where dominant strategies are absent. By employing randomised choices, players can maximise their expected probabilities of success. While randomising may seem counterintuitive, the application of game theory and empirical evidence from the literature demonstrate its effectiveness in real-world scenarios.

Back to the game. The shot was aimed at the left corner; the goalkeeper guessed correctly and got an outstretched hand to it, pushing it back into play, only to concede a goal on the rebound. Real Madrid got a chance to equalise from the spot eight minutes later and took it. And that’s how it ended at the Bernabeu. Real Madrid 1 Barcelona 1. Honours even!