Question

Let Row, Col, Layer be the players of the following 3-person, simultaneous game L1 L2 C1 C2 C1 C2 R1 (-4,-1,4) (-3,-4,2) R1 (1,-4,-4) (5,-5,-3) R2 (1,-2,-3) (-4,-3,-5) R2 (2,-1,-1) (0,4,2) R3 (1,-1,1) (2,2,3) R3 (0,5,-3) (2,1,2) (If simplex method is applied, write down the initial table (use column 1 as the pivot column) and the final table) (a) Find a prudential strategy for Row and his security level. (b) Find a prudential strategy for Col and his security level. (c) Find a prudential strategy for Layer and his security level. (d) Draw the movement diagrams for L1 and L2. Find all the saddle point(s) of the game. (e) Following Theorem 3 in Chapter 5, let $\vec{p} = (p_1, p_2, 1 - p_1 - p_2)$, $\vec{q} = (q, 1 - q)$, $\vec{r} = (r, 1 - r)$ be Row, Col, Layer's strategies respectively and $v_1, v_2, v_3$ be their payoffs respectively. The corresponding multilinear programming problem will then be Maximize ($\pi_1(\vec{p}, \vec{q}, \vec{r}) + \pi_2(\vec{p}, \vec{q}, \vec{r}) + \pi_3(\vec{p}, \vec{q}, \vec{r}) - v_1 - v_2 - v_3$) subject to $\vec{p}, \vec{q}, \vec{r} \ge 0$ together with 7 inequalities. Write them down explicitly. (f) For each of the saddle point(s) found in part (d), verify that it is a Nash equilib- rium by showing that it is an optimal solution of the multilinear programming problem in part (e).

          Let Row, Col, Layer be the players of the following 3-person, simultaneous game
L1
L2
C1
C2
C1
C2
R1 (-4,-1,4) (-3,-4,2)
R1 (1,-4,-4) (5,-5,-3)
R2 (1,-2,-3) (-4,-3,-5)
R2 (2,-1,-1) (0,4,2)
R3 (1,-1,1) (2,2,3)
R3 (0,5,-3) (2,1,2)
(If simplex method is applied, write down the initial table (use column 1 as the
pivot column) and the final table)
(a) Find a prudential strategy for Row and his security level.
(b) Find a prudential strategy for Col and his security level.
(c) Find a prudential strategy for Layer and his security level.
(d) Draw the movement diagrams for L1 and L2. Find all the saddle point(s) of
the game.
(e) Following Theorem 3 in Chapter 5, let
$\vec{p} = (p_1, p_2, 1 - p_1 - p_2)$, $\vec{q} = (q, 1 - q)$, $\vec{r} = (r, 1 - r)$
be Row, Col, Layer's strategies respectively and $v_1, v_2, v_3$ be their payoffs
respectively. The corresponding multilinear programming problem will then be
Maximize ($\pi_1(\vec{p}, \vec{q}, \vec{r}) + \pi_2(\vec{p}, \vec{q}, \vec{r}) + \pi_3(\vec{p}, \vec{q}, \vec{r}) - v_1 - v_2 - v_3$)
subject to $\vec{p}, \vec{q}, \vec{r} \ge 0$ together with 7 inequalities. Write them down explicitly.
(f) For each of the saddle point(s) found in part (d), verify that it is a Nash equilib-
rium by showing that it is an optimal solution of the multilinear programming
problem in part (e).

Let Row, Col, Layer be the players of the following 3-person, simultaneous game
L1
L2
C1
C2
C1
C2
R1 (-4,-1,4) (-3,-4,2)
R1 (1,-4,-4) (5,-5,-3)
R2 (1,-2,-3) (-4,-3,-5)
R2 (2,-1,-1) (0,4,2)
R3 (1,-1,1) (2,2,3)
R3 (0,5,-3) (2,1,2)
(If simplex method is applied, write down the initial table (use column 1 as the
pivot column) and the final table)
(a) Find a prudential strategy for Row and his security level.
(b) Find a prudential strategy for Col and his security level.
(c) Find a prudential strategy for Layer and his security level.
(d) Draw the movement diagrams for L1 and L2. Find all the saddle point(s) of
the game.
(e) Following Theorem 3 in Chapter 5, let
p⃗ = (p1, p2, 1 - p1 - p2), q⃗ = (q, 1 - q), r⃗ = (r, 1 - r)
be Row, Col, Layer's strategies respectively and v1, v2, v3 be their payoffs
respectively. The corresponding multilinear programming problem will then be
Maximize (π1(p⃗, q⃗, r⃗) + π2(p⃗, q⃗, r⃗) + π3(p⃗, q⃗, r⃗) - v1 - v2 - v3)
subject to p⃗, q⃗, r⃗≥ 0 together with 7 inequalities. Write them down explicitly.
(f) For each of the saddle point(s) found in part (d), verify that it is a Nash equilib-
rium by showing that it is an optimal solution of the multilinear programming
problem in part (e).

Added by Cynthia B.

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