L 2 - 4, ch 5-7, 11, 14, 15

Question 1

Best response

Answer 1

choosing the strategy that maximizes my expected payoff, given what I believe other players will do

Question 2

Basic rationality assumption

Answer 2

players do not play strategies that can be replaced by a better alternative, hence, rational players don’t play strictly dominated strategies

Question 3

A pure strategy si of player I is strictly dominated if

Answer 3

there is a strategy that has lower utility than all other strategies for all strategy profiles of other players

Question 4

In some games we might find champion strategies that are always better than anything else, called

Answer 4

strictly dominant strategies = a pure strategy of player I is strictly dominant if all other strategies have lower utility for all strategy profiles of the other players.

Question 5

Isomorphic games

Answer 5

(games giving the same incentives) – game may differ from each other by the strategic incentives they offer to the interacting players.

Question 6

Prisoner’s dilemma

Answer 6

is any 2 player 2 action game where each player has 1 strictly dominant strategy and the payoff outcomes of both players are lower when playing the payoff dominant strategy than when playing payoff dominated strategy.

Question 7

Common knowledge

Answer 7

a game is common knowledge if it holds for all players that A knows the game and A knows that B knows the game and B knows that A knows that B knows and so on. Hence, in a game with rational players, player A will never play strictly dominated strategy, and player B knows that and can expect that it won’t happen

Question 8

iterative elimination of strictly dominated strategies

Answer 8

The process of eliminating strictly dominant strategies. This works with some games, but not all. For instance in coordination game, no strategy is dominated by the other strategy.

Question 9

rationalizable strategies

Answer 9

The set of strategies that survives after iterated dominance

Question 10

RULE: In a finite game

Answer 10

there is at least one BR to every belief.

Question 11

Rational beliefs should not contain that

Answer 11

others play strictly dominated strategies

Question 12

In 2 player games there’s a direct link between

Answer 12

strict dominance and best responses.

Question 13

RULE: Every finite game has

Answer 13

At least 1 NE. No player can unilaterally increase own payoff by changing own NE strategy to some other strategy if all other players stick to the NE strategies

Question 14

Dominated strategy is never

Answer 14

a part of NE because it’s never BR.

Question 15

NE can involve players playing

Answer 15

a mixed strategy

Question 16

If player I plays a mixed strategy in a NE, then

Answer 16

all pure strategies of player I that are given a strictly positive weight in the NE must give to player I the same expected payoff against the NE strategies of the remaining players.

Question 17

How to find Pure Strategy NE in a matrix game:

Answer 17

1) Identify BR for each player

2) Identify strategy profiles where both players take BR.

Question 18

NE provides

Answer 18

Stability – no one has incentive to deviate.

Consistency – rational player takes a BR to a belief that others play the NE.

Question 19

Is NE efficient

Answer 19

Not always. It can lead to controversy - conflict between the individual and group rationality

Question 20

Is NE good for uniqueness and strategic uncertainty?

Answer 20

No, it’s often not unique so that our prediction based on it is not very sharp – the strategic uncertainty may prevail.

Question 21

A strategy profile is Pareto efficient if

Answer 21

there’s no profile that is more efficient. In Prisoner’s dilemma the NE is not Pareto optimal.

Question 22

Solving NFGs:

Answer 22

1. Minimal rationality requirement – delete strictly dominated strategies
2. When a game is common knowledge we can iteratively delete strictly dominated strategies
3. Identify BRs
4. NE is strategy profile where each player takes a BR

Question 23

NE in mixed and in pure strategies:

Answer 23

1. There always exist NE for finite games

2. Such existence can also be shown for other games (e.g. with strategy sets that are continuous closed intervals )

Question 24

EFG

Answer 24

extensive form game is based on an object called tree

Question 25

Tree

Answer 25

a collection of nodes and directed branches connecting these nodes that satisfy some conditions. Every tree has only one starting node (root). When we follow the direction of the arrows starting from it we can reach all final nodes and we never pass the same node twice – there are NO CIRCLES in trees.

Question 26

Tree rules:

Answer 26

1. Every node is a successor of the initial node, and the initial node is the only one with this property 2. Each node except the initial node has exactly one immediate predecessor. 3. Multiple branches extending from the same node have different action labels. 4. Each IS contains decision nodes for only one of the players 5. All nodes in a given IS must have the same number of immediate successors and they must have the same set of action labels on the branches leading to these successors.

Question 27

Circles, branches and final nodes denote

Answer 27

Circles (NODES) denotes the player who moves at that node Branches with arrows are the players possible actions Final nodes are the payoffs to all players when they end up at that node.

Question 28

We use IS to model

Answer 28

situations when a player doesn’t know what a player before him did. IS is a subset of nodes labeled by the same player which the player cannot distinguish from each other. IS are indicated by connecting the corresponding nodes of a player that belong to the same IS by a dotted line, or circling them together.

Question 29

2 nodes can only belong to the same IS if

Answer 29

the player deciding at the 2 nodes cannot possibly know his position in the game so: 1. All branches coming from a node in the IS have the same name 2. All nodes in the IS have the same amount of outgoing branches 3. All nodes in IS represent the same player

Question 30

Strategy is

Answer 30

a complete contingent plan of a player. In an EFG, strategy of a player has to specify what he would have done at each point a player might get to move. Hence, strategy is a collection of actions and there is one action/strategy specified for each IS of a player.

Question 31

2 EFGs can lead to

Answer 31

the same NFG, but for each EFG, there is only one NFG. Either player move sequentially but it doesn’t matter who moves first because no IS is given to the second mover OR the two players really move simultaneously.

Question 32

Ultimatum bargaining

Answer 32

the seller makes a single price offer p, take it or leave it offer.

Question 33

Information in a game is

Answer 33

A) Perfect if at the moment of choosing an action in a game, all players know all actions taken up to that point. B) Imperfect – any game that players move simultaneously C) Complete if all players know with certainty the payoffs all players obtain for each possible outcome of the game. Any game that players move simultaneously but they know the payoffs D) Incomplete – when there is uncertainty (modeled by Nature) about the payoffs (type) of some player(s)

Question 34

NFGs are

Answer 34

Complete and imperfect

Question 35

ESGs can be

Answer 35

both complete/incomplete and perfect/imperfect

Question 36

NEs of an ESG

Answer 36

these are the NE of the corresponding NFG

Question 37

Sequential rationality

Answer 37

An optimal strategy for a player should maximize his or her expected payoff, conditional on every IS at which this player has the move. That is, player i's strategy should specify an optimal action from each of player i's info sets, even those that player i doesn't believe will be reached in a game

Question 38

In games with perfect information we use

Answer 38

Backward induction

Question 39

In games with imperfect information we use

Answer 39

Subgame perfect NE SPNE

Question 40

Player’s strategy is sequentially rational if

Answer 40

it maximizes players expected payoff at each information set where the player has to move.

Question 41

When sequential rationality is common knowledge players can expect

Answer 41

that other players will not use strategies that are sequentially dominated. In practice, that means that they can use backward induction to identify the NE.

Question 42

Backwards induction outcomes are NE outcomes, since

Answer 42

they’re obtained by a process where at each step a player’s BR is chosen. -> Every finite game with perfect information has PSNE and BR identifies that NE.

Question 43

In games with imperfect information we implement the requirement of

Answer 43

subgame rationality via subgame perfection.

Question 44

Subgame of an EFG is

Answer 44

a collection of nodes of a game initiated by a single node such that any IS containing any nodes in the subgame is fully included in the subgame and all continuation nodes of any node in the subgame are also in the subgame. It is an EFG of its own. Each EFG contains at least 1 subgame that is the game itself. In a game of perfect info, every node initiates a subgame. A subgame starting at some other that the initial node is called proper subgame.

Question 45

A strategy profile is called SPNE if

Answer 45

it specifies a NE in every subgame of the original game.

Question 46

Most sports have a

Answer 46

matching-pennies component, where each side would be at a disadvantage if the other side knew its pure strategy. Randomization is thus an essential aspect of play.

Question 47

Duopoly with capacity constraints | Consider a variation of the Bertrand game

Answer 47

2 firms produce the same product at 0 cost and compete by selecting prices. There’re 10 consumers. Each consumer WTP = 1 /per unit. The firms simultaneously and independently select prices. In a setting with unconstrained capacities, all customers purchase from the firm that sets the lower price. If the firms charge the same part its 50% market share for each. However, let’s introduce capacity constraints. Each firm capacity = 8. If they both charge p1=p2, 5 consumers buy from firm 1 and 5 from firm 2. But if firm 1 charges a lower price, 8 consumers will purchase from firm 1 and the remaining from firm 2. Each firm has incentive to deviate from p1=p2, because it will give them 8 consumers and not 5. In sum, there’s no profile of prices that form a pure-strategy NE. Therefore, it must be mixed strategy. F (p1) = the probability that firm 2 price is below => p1 p2p1 Expected payoff of firm 1 = 8p1(1-F(p1)) + 2p1F(p1) F(p1) = (4p1 -1)/3p1 We know that F(p1) = 0 by definition. And Solving this we get p1 = ¼ In the mixed strategy profile the firms select prices by randomizing over interval [1/4, 1]. The randomization probabilities balance 2 forces: By raising its price, a firm increases its profit margin but decreases the probability that it will be the lower-priced firm and capture the higher quantity.