Chapter 12, Game theory II: Nonzero-sum and cooperative games Video Solutions, An Introduction to Decision Theory

Problem 1

Find all pure Nash equilibria in the following games.
$$
\begin{array}{llc}
\hline & \mathrm{C} 1 & \mathrm{C} 2 \\
\hline \text { R1 } & 3,5 & 0,4 \\
\text { R2 } & 1,0 & 20,30 \\
\hline
\end{array}
$$
$$
\begin{array}{lll}
\hline & \text { C1 } & \text { C2 } \\
\hline \text { R1 } & 0,8 & 2,2 \\
\text { R2 } & 8,0 & 3,3 \\
\hline
\end{array}
$$
$$
\begin{array}{llll}
\hline & \text { C1 } & \text { C2 } & \text { C3 } \\
\hline \text { R1 } & 0,2 & 2,2 & 2,3 \\
\text { R2 } & 4,4 & 3,3 & 1,2 \\
\text { R3 } & 3,2 & 1,3 & 0,2 \\
\hline
\end{array}
$$

Crystal Wang

Numerade Educator

Problem 1

A group of four people is about to select one out of six possible social states, $a, b, c, d, e$ or $f$. (a) Which state will be selected if the decision is based on a traditional voting procedure? (Does the result depend on how the voting procedure is set up, given that all people get one vote each?) (b) Which state would be selected by the maximin rule? (It prescribes that society should prefer a state in which the worst-off person is as well off as possible.)
Anne:$a>b>c>d>e$
Bert: $b>a>c>e>d$
Carl: $a>c>d>b>e$
Diana: $a>b>c>d>e$

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Problem 2

Consider the game below. It is a version of a coordination game known as 'the battle of the sexes'.
(a) What is a coordination game?
(b) Find the pure Nash equilibria of this game.
$$
\begin{array}{lll}
\hline & \text { C1 } & \text { C2 } \\
\hline \text { R1 } & 1,1 & 4,2 \\
\text { R2 } & 2,4 & 1,1 \\
\hline
\end{array}
$$

Lainey Roebuck

Numerade Educator

Problem 3

Recall Nash's solution to the bargaining game. One of the conditions he proposed for deriving his solution was the principle of irrelevant alternatives, holding that the solution to the bargaining problem does not depend on whether irrelevant strategies (i.e. strategies that are not preferred by any player) are removed or added to the set of alternative strategies. Can you construct a realistic scenario in which this condition is violated?

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Problem 4

Many games have more than one pair of equilibrium strategies. Does this show that game theory as we currently know it is an incomplete theory? Discuss.

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Problem 5

It is sometimes pointed out that cooperative games can in a certain sense be 'reduced to', or 'derived from', non-cooperative games. What does this mean? That is, what is the relation between cooperative and non-cooperative games?
(a) What is an iterated game?
(b) What does it mean to play tit for tat in the iterated prisoner's dilemma?
(a) What does the term 'evolutionary stable strategy' mean?
(b) Is evolutionary game theory a descriptive theory, or a normative theory, or something else?

James Kiss

Numerade Educator

04:50

Problem 8

Find all pure and mixed Nash equilibria in the following nonzero-sum games.pure and mixed Nash equilibria in the following nonzero-sum games.
(a) $$
\begin{array}{lrl}
\hline & \text { C1 } & \text { C2 } \\
\hline \text { R1 } & -3,-3 & 3,0 \\
\text { R2 } & 0,3 & 1,1 \\
\hline
\end{array}
$$
(b) $$
\begin{array}{lll}
\hline & \text { C1 } & \text { C2 } \\
\hline \text { R1 } & 0,3 & 4,2 \\
\text { R2 } & 6,2 & 3,3 \\
\hline
\end{array}
$$

Crystal Wang

Numerade Educator

01:50

Problem 9

Does evolutionary game theory presuppose that biological entities like animals and plants make rational decisions?

Ritika Shakkerwal

Numerade Educator

01:36

Problem 10

Which equilibrium concept do you think makes most sense, Nash's or the ESS concept used in evolutionary game theory? Does it make sense to maintain that different equilibrium concepts should be used in different contexts? Why/why not? Discuss!

Andrew Davis

Numerade Educator

Problem 11

Many philosophers have attempted to establish a link between game theory and ethics. How can such attempts overcome Hume's law, according to which no moral conclusions can be derived from purely non-moral premises?

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