Week 3 - Game Theory - The basics Flashcards
What are the 4 different games with different levels of information?
- Static games of complete information
- Dynamic games of complete information
- Static games of incomplete information
- Dynamic games of incomplete information
What are the 5 rules every game is played by?
- Who is playing (which players)
- What they are playing with (alternative actions or choices)
- When each player gets to play (or in what order)
- How much they stand to gain
- What each player knows when they act
What are the Cartesian products of 2 sets?
The Cartesian product of 2 sets AXB is the set of all possible ordered pairs
What are the 4 essential concepts of game theory?
- Best Response
- Dominance
- Nash Equilibrium
- Pareto efficiency
What is the definition of the best response concept?
The best response is the strategy or strategies which produce the most favorable outcome for a given player, taking as given the strategies of other players.
What is a dominance strategy and a dominated strategy?
- Dominance occurs when one strategy is better than another regardless of the other player’s strategies.
- A dominated strategy is one in which you are better off not playing at all because you will be worse off and would not rather play.
What is the Nash equilibrium?
- A Nash equilibrium occurs when a player is best responding given the actions of the other players
What is a pure strategy and a mixed strategy?
- Pure strategy is a strategy you play with 100% certainty e.g confess
- Mixed strategy is when you are randomized across several pure strategies e.g playing confess with 50% and not confess with 50%
Explain the rock paper scissors game in terms of best response, dominant strategy, and Nash equilibrium
- No unique best response as it all depends on the other person
- No dominant strategy as we don’t know what the other person picks which could make whatever we potentially pick bad.
- No Nash equilibrium, because there is no unique best response so individuals cannot choose a strategy in which they are best responding
Define Pareto Efficiency
- An outcome is said to be Pareto efficient if its impossible to so that one person is better off and the others are no worse off
- Therefore Pareto efficiency is the point where changing one thing will make someone else happier and the others less happy.
In a prisoners dilemma explain which is the dominated strategy and if there is a Pareto efficiency
- Imagine Player X → and player y downwards. With confess first then not confess
- Confess is a dominant strategy for Y as whatever X picks the best response for Y would confess. For example, if X picks confess the best option is confess, and if X picks not confess the best option is confess.
- Y not confessing and X confessing 1,-2, is Pareto efficient as moving from this point will make someone better off but at the same time someone worse off.
What is the Nash equilibrium for mixed strategies?
- Let J denote the number of pure strategies in S1, and let K denote the number in S2.
- We have S1 = {s11,…….s1} and S2 = {s21,…..,s2k]
- If player 1 believes that player 2 will play the strategies {S21,….,S2K} with probabilities (P21,…..,P2K) then players 1’s expected pay off from playing the pure strategy s1j is: Sum(k=1,K) P2kU1(s1j,s2k)
What is the expected Pay off to player 1 for playing the mixed strategy p1=(p11,…,p1j)?
V1(p1,p2) = Sum (J,j=1) P1j [ Sum (K,K=1) p2kU1(s1j,s2k)]
If player 2 believes that player 1 plays the strategies {s11,….,s1j} with probabilities (p11,…,p1j) the expected Pay off to player 2 is from playing the mixed strategy is (p2,1…..,p2k) is?
v2(P1,P2) = Sum (K,k=1) p2k [Sum J(j=1) p1ju2(s1j,s2k)]
Define the Nash equilibrium with mixed strategies.
- In a 2 player normal-form game, the mixed strategies (p1,p2) are a Nash equilibrium if each player’s mixed strategy is the best response to the other player’s mixed strategy. The 2 conditions below must hold
- V1(P1,P2) ≥ V1(P1,P*1)
- V2(P1,P2)≥V2(P*1,P2)
