Game Theory
Dominant strategies will
always be better than all other strategies
Dominated strategies will
always be worse than all other strategies
First-mover advantage comes from ...
an advantageous position and be able to force the other players to adapt to it.
Whats the profit function?
(Px-Cx)* Qx
How do we classify games?
-Timing of play -Common or conflicting interest -Number of times an interaction occurs -The amount of information available to each players -The type of rules -Feasibility of coordinated action
What is a strategic game?
-Two or more players. -Each player responds to what the other will do because moves by other players affect your own payoffs -Mutual Awareness --Non-random ("strategic") Actions
How do we know when we have a Prisoners Dilemma?
1.) It looks like this. 5,5 .... 12,2 2,12 ... 11,11 1. IN GENERAL, prisoner's dilemmas will have TWO players with TWO strategies each. 2. Those strategies will fall into two categories, which can be thought of generally representing "cooperate" and "don't cooperate" type moves, in regards to the other player. IE, For prisoners, "cooperating" means cooperating with the other prisoner by "NOT CONFESSING," but for prisoner's dilemmas without prisoners, "cooperate" still refers to cooperation with the other player, in order to attain the best outcome. 3. Furthermore, the game should be simultaneous and not allow for communication between players 4. MOST IMPORTANTLY, you need to draw the game out and solve it. What you should see when you do this is ONE NASH SOLUTION in one corner, and then diagonally across from that, a set of payoffs WHICH ARE BETTER FOR BOTH PLAYERS than the payoffs of the Nash. WITHIN each of these two cells, the payoffs should be equal. 5. In the other two boxes, you should see two unequal payoffs, which are switched diagonally across the cells. 6. In each outcome with unequal payoffs, there should be a HIGHER PAYOFF for the player who DOESN'T COOPERATE (and in fact, this payoff should be the BEST available payoff in the whole game), and a LOWER PAYOFF for the player who COOPERATES (and this payoff should be the WORST payoff available out of the entire game). *the Nash you find should take the form of {Don't Cooperate, Don't Cooperate}, while the cell diagonal from that (which has better payoffs for both players than the Nash solution), should take the form of {Cooperate, Cooperate}, if the game IS in fact a prisoner's dilemma.*
What is a total conflict game?
A zero sum game.
What kind of game is this? Angelina NY LA Brad NY (4, 2) (0,0) LA (0,0) (2,4)
Battle of the sexes game
What kind of game is this? 2,1....0,0 0,0....1,2
Battle of the sexes!
"A game player would never prefer an outcome in which every player gets a little profit to an outcome in which he gets all the available profit." True or False?
FALSE!
If the iterated elimination of dominated strategies method does not determine a Nash Equilibrium, then the game has no Nash Equilibrium. True or False?
FALSE!
In a sequential game, the player moving first has always an advantage. True or False?
FALSE!
If the best response analysis method does not determine a Nash Equilibrium, then the game has no Nash Equilibrium. True or False?
FALSE! Two things to remember: Every game has a Nash in mixed strategies, and if best response doesn't find the Nash, its in mixed strategy.
A simultaneous-move game always has at least one Nash. True or False?
False
If a game has a NE in mixed strategy, then the game has no equilibrium in pure strategy. True or False
False
If a game has no NE in mixed strategy, then the game has a NE in pure strategy. True or False
False
If a game has several NE in pure strategy, then the game has no NE in mixed strategy. True or False?
False
If iterated elimination of dominated strategies method does not determine a Nash, then the game has no Nash. True or False?
False
If the best-response analysis method does not determine a Nash Equilibrium, then the game has no Nash Equilibrium. True or False?
False!
If the iterated elimination of dominated strategies method does not determine a Nash Equilibrium, then the game has no Nash Equilibrium. True or False?
False!
The minimax method can be used to find the Nash Equilibrium in any type of simultaneous move games. True or False?
False!
If the best response analysis method does not determine a Nash Equilibrium, the the game has no Nash. True or False?
False. It might have one in Mixed strategies
The New York Times and The Wall Street Journal choosing the prices for their online subscriptions. Game or Decision?
GAME.
A game has a second-mover advantage if
considering a hypothetical choice between moving first and second, a player would choose the latter.
First-mover advantage ...
does not exists in all games.
A one shot game ...
means there no need to build a reputation and there's no possibility of punishing a player. An example of this is tourist.
Second-mover advantage comes from...
the flexibility to adapt oneself to the other's choices.
Russia choosing a level of tariffs to apply to European imports. Game or Decision
Game
Two students are working together on a project. Each player chooses the level of implication and effort to put into the project.
Game
What is an example of a strategic game?
Golden Ball : 2 players, Actions: Split or Steal, , Mutual awareness of the cross effect of their actions.
If each player isn't being selfish, then does that mean all players have the same value?
NO! An example of this is Donald Trump and Obama
Most economic and Social games are ...
Non Zero Sum games
What is a partial coordination game?
Non-Zero Sum Games.
Four roommates, each owning a personal computer in their rooms, decide to decrease their electricity bill. To achieve their goal, they all agree to reduce the use of their Personal Computer by 20%. The four roommates face a ...
Non-cooperative game
Ten players play the following game: Each participant choose simultaneously a number between 0 and 100. Then a judge computes the average. The winner is the person whose choice is closest to 1/2 of the average. The Nash equilibrium of the game is
Zero.
In some sequential games, the player moving first has an advantage. True or False?
True.
Whats an example of asymmetric information?
Used Cars
In a game like this: What do you use to solve? 2,2 .....0,0 0,0......2,2
We use the Best Response because we can't use iterated dominance or dominant strategy.
Best Response *IN PURE STRATEGIES* when doing equations is ...
X=B/2C *Remember that "X" here is really going to be either "Px" or "Qx"*
Maximization of functions ?
Y = A + BX - Cx^2 *remember that "Y" here is really going to be either "Px" or "Qx"*
What is a simultaneous game?
Your current move is governed by what you think your opponent is doing right now. You must square the circle of "he thinks that I think..."
Whats a sequential game?
Your current move is governed by your calculation of its future consequences. You must anticipate what the actions of your opponent will be.
Best response analysis will always find all Nash equilibria
in *pure strategies*, *IF THEY EXIST*. If there is no Nash in pure strategies, Best Response isn't going to give you any Nash in pure strategies.
Simultaneous means
information set
Discrete strategy means
limited number of strategies.
A NE can be defined as a
list of strategies, one for each player, such that no player can get a better payoff by switching to some other strategy that is available to her while all other players to adhere to their own listed strategy.
Chess is an exmaple of a game with
perfect information.
The term Cournot competition refers to a situation where firms compete in
quantity.
Game theory is the study of
rational behavior is interactive situations.
A sub-game is
A game compromising a portion or remnant of a larger game, starting at a non-initial node of the larger game.
A Nash Equilibrium can be defined as:
A list of strategies, one for each player, such that no player can get a better payoff by switching to some other strategy that is available to her while all other players adhere to their listed strategy.
A Nash equilibrium can be defined as ...
A list of strategies, one for each player, such that no player can get a better payoff by switching to some other strategy that is available to her while all other players adhere to their listed strategy.
An information set is
A set of nodes among which a player is unable to distinguish when taking an action.
A subgame-perfect equilibrium (SPE) is
A set of strategies such that their continuation in any subgame remains optimal, whether that subgame is on or off the equilibrium path.
What is an uncooperative game?
Agreements are not enforceable, actions are not observable, and no party can enforce compliance. Cooperation will arise if and only if it is in the participant's separate and individual interests to comply
Which affirmation is true?
All SPE are Nash equilibria
What kind of game is this? Angelina NY LA Brad NY (2, 2) (0,0) LA (0,0) (1,1)
Assurance game
What are examples of zero sum games?
Chess, basketball, Soccer, Tennis
What kind of game is this? Angelina NY LA Brad NY (-2,-2) (-1,1) LA (1,-1) (0,0)
Chicken game
Whats an example of a zero sum game with three or more players?
Cycling.
Jim Jam receives a gift card to buy "The Life of Pi" and must decide to buy the e-book or the paper version. Game or Decision?
DECISIONS.
Evan has a midterm in Game Theory today. He has to choose the level of effort to put into the midterm. Game or Decision.
Decision
When a consumer rent a car at Avis and choose the "full coverage" option, the consumer and Avis play a game with Imperfect or Perfect information?
Imperfect Information
When you buy a car, you play a game with...
Imperfect Information.
Tennis is....Zero-sum game or not a zero-sum game?
It's a ZERO-SUM GAME!
What are cooperative games ?
Its a game where agreements are enforceable, actions are observable, and third party can enforce compliance.
What is an example of a decision game?
Lottery Golden Ball : 2 Players, Players can't look at whats on the inside of the balls, Actions: Player chooses a ball, Players can't respond to what the other does. Basically it's PURE CHANCE.
Which affirmation is true
Mixed strategies can be used in both zero-sum games and non-zero sum games.
In game theory are all players selfish?
NO! An example of this is Bill Gates and Warren Buffet.
In game theory, "discrete strategies" means...
Players have a limited number of strategies.
In game theory, to have "continuous strategies" means
Players have nearly an unlimited number of strategies available.
If a first/second-mover advantage exists and rules are manipulable then...
Pregame is as important as the game.
If you find two Nash Equilibrium's, it is not a
Prisoners Dilemma.
How do we find expected payoff?
Probability times payoff added together
∏x = (Px-Cx)*Qx
Profit Function
Whats the difference between pure coordination and assurance game?
Pure Coordination 2,2 .....0,0 0,0......2,2 Assurance Game 5,5.......0,0 0,0.......1,1 -Pure coordination is just a matter of the two players talking and meeting up at the same place cause they don't care either way. Think of them as being in a relationship. - In an assurance game, players actually have the SAME PREFERENCE for where they would rather be.Think of this as two people in a relationship, but one broke their phone--they can either meet up for dinner at the Hub (sad face) or go to that nice Trattoria place off campus. The only reasons they would go to the Hub over Trattoria is that they want to eat and be together, and also one or both of them is too tired to walk to Trattoria. Thus, although they'd BOTH rather eat food from not-the-Hub, the Hub is also a Nash since one or both of them might be too tired to walk off campus.
What kind of game is this? 2,2 .....0,0 0,0......2,2
Pure Coordination !
For a repeated game
Reputation is crucial. Punishing might exist. Example, the neighbor of a restaurant.
Sealed-bid auction: Bidders on bottle of wine seal their bids in envelopes. The highest bidder wins the item and pays the amount of his bid. Sequential or Simultaneous ?
SIMULTANEOUS.
An English auction: The auctioneer opens the auction by announcing a Suggested Opening Bid ( a starting price) for item and then accepts increasingly higher bids from the floor consisting of potential buyers. The highest bidder at any given moment is considered to have the standing bid, which can only be displaced by a higher bid from a competing buyer. If no competing bidder challenges the standing bid within a given time frame, the standing bid becomes the winner. Simultaneous or Sequential?
Sequential.
In some sequential games, the player moving second has an advantage. True or False?
TRUE!
The Minimax method can be used to find the Nash Equilibrium only in the case of a zero sum game. True or False?
TRUE!
Whats an example of a simultaneous game and a sequential game?
Tablets with apple vs. Microsoft vs kindle
Whats an example of a cooperative game?
The lease between the landlord and tenant.
A mixed strategy for a player consists of
a random choice, to be made with specified probabilities, from his originally specified pure strategies.
Iterated elimination of dominated strategies and dominant strategy won't always or successfully
find ALL the Nash equilibria, in pure strategies, even if they exist. Best response analysis on the other hand, will always find all Nashes in pure strategies, if they exist.
Complete the following sentence: A game has perfect information if all of its information sets consists of singleton nodes.
perfect
One important characteristic of a "prisoners dilemma" is that
the dominance solution equilibrium is worse for both players than the non-equilibrium situation in which each plays the dominated strategy.
One of the important characteristic of a "prisoner's dilemma" is that
the dominance solution equilibrium is worse for both players than the non-equilibrium situation in which each plays the dominated strategy.
One important characteristic of a "prisoners dilemma" is that
the dominant solution equilibrium is worse for both players than the non-equilibrium situation in which each plays the dominated strategy.
The opponents indifference property states that
the equilibrium in mixed strategy satisfies the condition that the other player should be indifferent between her two strategies.
Game Theory is
the science of rational behavior in interactive situations.
If you can answer who moves first
there shouldn't be an information set.
In a pure coordination game there are
two Nash Equilibrium.
