SMU H3 Map

• Content map: SMU H3 Game Theory Map

Mixed Strategy Recap

Step 1

• Write and solve the two inequalities of which $A$ gives the biggest expected payoff for P1 in terms of P2’s probabilities

• Write and solve the two inequalities of which $B$ gives the biggest expected payoff for P1 in terms of P2’s probabilities

• Write and solve the two inequalities of which $C$ gives the biggest expected payoff for P1 in terms of P2’s probabilities

Step 2

• Repeat for expected payoff for P2 in terms of P1’s probabilities

Step 3

• Solve for inequalities and draw the best response graph

Fully Mixed Equilibria

• At the point where all three inequalities meet, it represents the probabilities where the player is indifferent to all three choices (i.e. payoffs are the same)
• This represents the fully-mixed Nash equilibrium

Partially Mixed Equilibria

• Along the line of intersection of two regions, it represents the probabilities where the player is indifferent to both choices (i.e. payoffs are the same)
• At the same time, the player is worse off from choosing the last option not in the bag
• This represents the partially-mixed Nash equilibrium

Repeated Games

Conditions

• Game has an uncertain end, and it is possible to infinitely repeat the game
• You must care about the future impact and not be myopic

Definitions

• A repeated game is a supergame where the same game (“stage game”) is played for a certain number of rounds in succession, $T$

  • $T$ can be finite (finitely repeated game), or
  • $T$ can be infinite (infinitely repeated game)

• The payoffs are the sum of payoffs earned a each round

• The information is such that the players know the outcome of each game at each round

• The strategies of the game can be represented as the following:

  • Let the history of the game be the outcomes from round $1$ to round $t$ be $h_t$
  • The strategy is a rule mapping any history $h_t$ into a choice of the base game
  • Strategies are a complete descripton of the choies that the player will make, so they must prescribe an aciton at each round for any possible history

Finite Game

Example

• Prisoners’ Dilemma repeated twice ($t=2$)
• Since the first round has 4 possible outcomes, the strategy for a player must tell what choice to make at:
  • Round 1: ($C$ or $D$)
  • Round 2: choice to make in any of the 4 possible outcomes ($CC$, $CD$, $DC$, $DD$)

Strategies

Grim-Trigger:

• Start with $C$
• As soon as one player chooses $D$, play $D$ forever

Tit-for-Tat

• Start with $C$
• Mimic the other opponents’ choice the previous round $t-1$

Unconditional Cooperator

• Always play $C$

Unconditional Defector

• Always play $D$

Constricting the Sub-game Perfect Equilibrium

• For a two-round prisoner dilemma
• Starting at round 2, since it is the last round, there is a dominant strategy for each player ($D$), the equilibrium is $\{D, D\}$ regardless of subgame
• Move the payoffs from the subgames of round 2 to attain the following payoff matrix

• Since players are incentivised to defect when choosing $\{C, C\}$, or $\{C, D\}$ or $\{D, C\}$
• The best response for either player is $D$, meaning that the Nash equilibrium is $\{D, D\}$
• Unconditional defection is the subgame perfect equilibrium

Infinite Game

Think

• Observation 1:
  • Since there is no end, backward induction is not feasible
  • However, you can create recursive systems which allow for effective calculation
• Observation 2:
  • You cannot look at payoffs in the same way since comparing infinite payoffs with infinite payoffs does not make sense
  • There is a decay which comes with time since we value completing things now than latter

Example 1

• Players are impatient, so $1 today is worth more than $1 later
• Suppose you deposit $D$
  • After 1 year you have $D + \gamma D = D(1 + \gamma)$
  • After 2 years you have $D(1 + \gamma) + \gamma D(1 + \gamma) = D(1 + \gamma)^2$
  • After 3 years you have $D(1 + \gamma)^2 + \gamma D(1 + \gamma)^2 = D(1 + \gamma)^3$

Example 2

• If you want to return $P$ some time from now
• The present value of \ P $ one year can be represented as

• The present value of \ P $ two years can be represented as

• The present value of \ P $$t$ years can be represented as

• Let $\delta = \frac{1}{1+\gamma}$, we have that

Application to Game Theory

• Suppose the sequence of payoffs for P1 is

• The payoff in the repeated game will be

• We consider round $1$ as today, and we discount future payoffs on the basis of how far they are from round 1

Another Example 3

• If players always cooperate, P1 earns 4 in each round

Best Response strategy

• Let us call the infinitely repeated game $\mathcal G$
• Let $\sigma$ denote a strategy for $\mathcal G$, where $(\sigma, \sigma)$ is a subgame perfect equilibrium if and only if no player can gain by deviating just once at every subgame
• Deviating will bring the game into another subgame state

Example

• Grim-trigger is a subgame perfect equilibrium if $\delta$ is large enough

• This means players are not too impatient or myopic

• With grim-trigger, we have two possible families of subgames

  1. At least one player chose $D$
  2. No $D$ chosen so far

• If history 1, play $C$ for one round

• If history 2, play $D$

• In history 1, if players stck to grim-trigger, P1 earns (moving forward)