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Dynamic Noncooperative Game Theory

Tamer BaÅŸar and Geert Jan Olsder

Chapter 2

Noncooperative Finite Games: Two-person Zero-sum - all with Video Answers

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Chapter Questions

Problem 1

Obtain the security strategies and the security levels of the players in the following matrix games. Which one(s) admit (pure) saddle-point solutions?

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Problem 2

Obtain the mixed security strategies and the average security levels of the players in the following matrix games.

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Problem 3

Verify that the quantity $\sup _Z \min _Y y^{\prime} A z$ is unique.

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Problem 4

Prove that the quantity $\max _z y^{\prime} A z$ is a continuous function of $y \in Y$.

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Problem 5

Determine graphically the mixed saddle-point solutions of the following matrix games.

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Problem 6

Convert the following matrix games into linear programming problems and thereby numerically evaluate their saddle-point solutions.

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04:51

Problem 7

The space of all real $(m \times n)$ matrices $A=\left\{a_{i j}\right\}$ can be considered to be isomorphic to the $m n$-dimensional Euclidean space $\mathbf{R}^{m n}$. Let $S_{m n}$ denote the set of all real ( $m \times n$ ) matrices which, considered as the pay-off matrix of a zero-sum game, admit pure strategy saddle-point equilibria. Then $S_{m n}$ is isomorphic to a subset of $\mathbf{R}^{m n}$, to be denoted by $S^{m, n}$.
(i) Is $S^{m, n}$ a subspace?
(ii) Is $S^{m, n}$ closed, convex?
(iii) Describe $S^{2,3}$ explicitly.

Hunza Gilgit
Hunza Gilgit
Numerade Educator

Problem 8

Let $T^i(A)$ denote the set of all mixed saddle-point strategies of $\mathbf{P} i$ in a matrix game $A$. Show that $T^i(A)$ is nonempty, convex, closed and bounded.

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02:02

Problem 9

A matrix game $A$, where $A$ is a skew-symmetric matrix, is called a symmetric game. For such a matrix game prove that $T^1(A)=T^2(A)$, and $V_m(A)=0$.

Nez Nikoo
Nez Nikoo
Numerade Educator

Problem 10

Obtain the pure or behavioural saddle-point solutions of the following single-act games in extensive form.

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Problem 11

Obtain a feedback saddle-point solution for the feedback game in extensive form depicted in Fig. 13. What is the value of the game? What is the actual play dictated by the feedback saddle-point solution? Show that these constant strategies also constitute a saddle-point solution for the feedback game.

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Problem 12

Show that the feedback game in extensive form depicted in Fig. 14 does not admit a pure-strategy feedback saddle-point solution. Obtain its behavioural saddle-point solution and the behavioural saddle-point value. Compare the latter with the value of the game of Problem 11. What is the actual play dictated by the feedback saddle-point solution?

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Problem 13

What is the open-loop version of the feedback game of Fig. 14 ? Obtain its saddle-point solution and value.

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Problem 14

Prove that if a feedback game in extensive form admits a feedback saddlepoint solution with the value $J_f$, and its open-loop version (if it exists) admits a unique saddle-point solution with the value $J_0$, then $J_f=J_0$. Furthermore, show that the open-loop saddle-point solution is the actual play dictated by the feedback saddle-point solution.

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Problem 15

Verify the following conjecture: If a feedback game in extensive form admits a behavioural saddle-point solution which actually dictates a random choice at least at one level of play, then its open-loop version (if it exists) cannot admit a pure-strategy saddle-point equilibrium.

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Problem 16

Investigate whether the following multi-act game of the delayed commitment type admits a behavioural saddle-point solution or not.

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Problem 17

Determine the pure or behavioural saddle-point solution of the following game in extensive form.

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Problem 18

Determine the pure or behavioural saddle-point solution of the following game in extensive form which also incorporates a chance move.

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Problem 19

Determine the solution of the previous problem in the case when $\mathbf{P} 2$ has access to nature's choice. What is the value of this extra information for P2?

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01:54

Problem 20

Construct an example of a matrix with only real eigenvalues, which, when considered as a zero-sum game, has a value in mixed or pure strategies that is smaller than the minimum eigenvalue.

Gennady Notowidigdo
Gennady Notowidigdo
Numerade Educator

Problem 21

A matrix game is said to be completely mixed if it admits a mixed saddlepoint solution which assigns positive probability weight to all pure strategies of the players. Now, if $A=\left\{a_{i j}\right\}$ is a nonsingular ( $n \times n$ ) matrix with nonnegative entries, prove that the game whose pay-off matrix is $A^{-1}$ is completely mixed, with average value $V_m\left(A^{-1}\right)=1 /\left(\sum_{i=1}^n\right.$ $\sum_{j=1}^n a_{i j}$ ).

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