Overview of mixed strategies and evolutionary games week 2

Question 1

what are the five parameters that define game theoretic models?

Answer 1

1. The players: A player is an entity that makes decisions; models describe the number and identities of players.
2. The feasible strategy set: Actions with a positive probability of occurring
3. The outcomes or consequences: the feasible strategies of all players intersect to define an outcome matrix
4. The payoffs: Every outcome has a defined payoff for every player. Players are assumed to be rational, that is, to prefer a higher payoff to a lower one.
5. The order of play: Play may be simultaneous or non-simultaneous, i.e., sequential.

Question 2

Pure strategy definition

Answer 2

A strategy in which a player makes a specific choice or takes specific action

Question 3

Mixed strategy definition

Answer 3

A strategy in which a player makes a random choice among two or more possible actions, based on a set of chosen probabilities. (Their behaviour is unpredictable)

Question 4

Mixed strategy Nash Equilibrium (MSNE) definition

Answer 4

A mixed strategy nash equilibrium (MSNE) is one in which players randomise optimally - they are doing the best they can given the other is randomising

Question 5

EPO meaning

Answer 5

Expected Payoff

Question 6

APO meaning

Answer 6

Average Payoff

Question 7

Characteristics of Evolutionary Games

Answer 7

• Origins in biology for survival (survival of the fittest)

If some types are fitter than others:
- The fittest have greater reproductive success - become more numerous
- Less fit become less numerous - survival threatened

• This is not an evolutionary stable situation (This situation, where one type dominates entirely, is not stable. Over time, new strategies or mutations may appear and disrupt the balance).

Question 8

What conditions are needed for a population to be stable in evolutionary games (hawks and doves example)

Answer 8

1. No type has a higher APO (average payoff). If one has a higher APO then their reproductive rate is higher - the population mix will change
2. Not Vulnerable to Invasion:
   Stability also requires that no invading strategy (e.g., a few hawks entering a mostly dove population) can outperform the existing population’s strategies. If the invaders perform better, they will grow in number, disrupting stability.