L9 - Game Theory: Mixed Strategies I

Question 1

Examples of real-world games that need a mixed strategy?

Answer 1

• Aside from sports and games (where there may be always an incentive to deviation and thus no Nash equilibrium), there are some related important real-world applications:
  
  • Pricing and advertising
  • Terrorism and crime
  • Inspecting and auditing
  • Military

Question 2

What is a mixed strategy Nash equilibria?

Answer 2

• This immediately suggests that some form of randomisation is likely to occur within the game – players want to be unpredictable, otherwise, they will be outmanoeuvred by their rivals.
• More formally, these sorts of games have no Nash Equilibrium in pure strategies.
• However, we will now show that there exists a different style of Nash Equilibrium – in mixed strategies.
  
  • Pure Strategy: Where a player selects a single strategy deterministically.
  • Mixed Strategy: Where a player selects their chosen strategy from a probability distribution
    
    • . E.g. In this context, a mixed strategy could involve playing Left and Right with probability p and (1-p) respectively
• In the case of mixed strategies, the same Nash equilibrium principle can apply:
  
  • In eqm, given the behaviour of the other players, each player has no strict incentive to change their strategy (or probability distribution over strategies).
  • Once one allows for mixed-strategy equilibria, it can then be shown that every possible game has at least one Nash Equilibrium under some mild regularity conditions!

Question 3

How can we use a mixed strategy to solve the football game?

Answer 3

• for the rock paper scissors game –> equilibrium is where each player randomises across the three strategies with a probability 1/3

Question 4

What is the typical situation when mixed strategy equilibria will arise?

Answer 4

• in the second case their are two pure strategies but also a mixed strategy equilibria aswell t

Question 5

What are some cons of mixed strategy equilibrium?

Answer 5

• Mixed strategy equilibria are subtle, complex, and often quite strange. Some would go further and say they are unrealistic. economist vary in their views of their validity:
  
  • • Do economic agents really randomise their decisions? Are agents ever truly indifferent between strategies? –> how often do we just randomise what I want for dinner –> never perfectly indifferent always have some reason to justify your choice
  • • More than that, is it reasonable to require players to randomise using a specific distribution in order to keep their opponent indifferent? –> when do we ever think like that
  • • By definition, mixed strategy equilibria are not very stable - players exhibit ex-post regret: in hindsight, they would sometimes change their behaviour –> wish I went the other way on a penalty saved
  • • Some people may not have the capacity to randomise effectively. If not, then cleverer players may exploit any predictability:
    
    • o Bart and Lisa Simpson playing Rock-Paper-Scissor
    • o Derren Brown playing Rock-Paper-Scissor

Question 6

What are the Pros of Mixed Strategy Equilibria?

Answer 6

• Help analyse situations where pure eqm analysis would otherwise fail –> understand why their wouldn’t be a pure eqm
• Help capture an intuitive feature of unpredictability
• • Aside from sports, and poker, applicable to many areas of economics: inspecting+auditing, crime+terrorism, pricing, advertising, … Also, relevant for military applications, e.g. D-Day landing destination.
• • (There are also some higher-level justifications for their existence that are not included on this module: e.g. evolutionary or purification justifications.)

Question 7

What is the evidence that supports mixed strategy equilibria?

Answer 7

• Lab experiments show some limited support, individuals often come close to the mixed eqm but with some significant departures, e.g. players often alternate across their strategies too frequently relative to that predicted.
• However, evidence from other real-world settings can be stronger. For instance, when individuals are experienced and when the stakes are high as in professional sport, the evidence for mixed Nash equilibria is strong.
• The success rates from different strategies are often equal (e.g. scoring rates for left and right penalties in football and point winning rates for left and right serves in tennis (Palacios-Heurta 2003 and Walker and Wooders 2001)).
  
  • Could be argue they have more incentives than those in laboratory experiments as they have less incentives to practice and employ such strategies

Question 8

When are mixed strategies useful?

Answer 8

• Have a certain appeal when players are tyryin to be unpredictable as in games of pure conflict (penalty taking)
• They also make sense when one player is trying to deter some action by another for instance in quality control exercises in which auditors make random checks to discourage malpractice
  
  • Random checks as it would be hard to check anything
• Sometimes rational to do when a player just doesn’t known what else to do
• It should be noted that mixed strategies won’t always ake sense even if neither player has a dominant strategy
  
  • Battle of the sexes both rational players really want to coordinate even though they both have a different preference –> may look for alternatives to mixed strategy like committing or looking for a focal point (focal point in game theory is an outcome that stands out in some way for the player to coordinate their actions for0