Game Theory Basics 1st Edition Bernhard Von Stengel - Solutions

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(a) Draw the best response diagram for this game.(b) Identify all equilibria of this game.(c) Write down the Lemke-Howson path that starts at \((\mathbf{0}, \mathbf{0})\) for each missing label \(1,2,3,4,5,6\), as in (9.7), by suitably annotating the nodes in \(\tilde{X}\) and \(\tilde{Y}\). Which
Consider the degenerate \(2 \times 3\) game in Figure 6.12. Construct the best response diagram (note that \(X\) is only a line segment) and identify the set of all equilibria with the help of the labels. II 1 I T payoff to II 2 2 2 1 0 1 0 0 B 3 2 1 2 C 1 1 2 0 0 0 12 prob(B) 1 Figure 6.12
Consider the following \(3 \times 3\) game.(a) Find all pure strategy equilibria of this game.(b) The picture below shows, for this game, the mixed-strategy sets \(X\) and \(Y\) for players I and II, but not which set is \(X\) and which is \(Y\). It also shows their subdivisions into best-response
Consider a nondegenerate bimatrix game and an arbitrary purestrategy equilibrium of the game (assuming one exists). Explain why the LemkeHowson path started at \((\mathbf{0}, \mathbf{0})\) always finds this equilibrium for a suitable missing label. Suppose the game has at least two pure equilibria.
Give an example of a bimatrix game \((A, B)\) where every column of \(A\) has a positive entry but where \(Q\) in (9.24) is not a polytope.
Consider a nondegenerate bimatrix game that has a pure-strategy equilibrium. Show that there is a strategically equivalent game so that in this pure equilibrium, both players get a strictly higher payoff than for any other pair of pure strategies. Explain why the "dilemma" in the Prisoner's Dilemma
Explain the two ways, with either player I or player II moving first, of representing an \(m \times n\) bimatrix game as an extensive game that has the given game as its strategic form. Giving an example may be helpful. In both cases, how many decision nodes do the players have? What is the number
Which of the following extensive games have perfect recall and if not, why not? For each extensive game with perfect recall, find all its equilibria in mixed (including pure) strategies. a 55 L II T R I T a S 9 II 1 L a b II 4 -1 -4 d -5 505 1 -1 -2 NN 2 3 -3 I R II a C d 90 S 4 ++ -4 S 1 -1 T 2 3
Which of the following extensive games have perfect recall and if not, why not? For each extensive game with perfect recall, find all its equilibria in mixed (including pure) strategies. L R a b II II a b P 0 70 69 NO 5 2 1 0 30 L II R S T 30 a a b 2 70 69 2 NO 51
Consider the following zero-sum game, a simplified version of Poker adapted from Kuhn (1950). A deck has three cards (of rank High, Middle, and Low), and each player is dealt a card. All deals are equally likely, and of course the players get different cards. A player does not know the card dealt
For an extensive game with perfect recall, show that the information set that contains the root (starting node) of any subgame is always a singleton (that is, contains only that node).
Find all equilibria in (pure or) mixed strategies of this extensive game. Which of them are subgame-perfect? a II I T B T B 1 1 4 5 II d II e 55 5 5 22 1 6 2 1 4
Consider this three-player game in extensive form. At a terminal node, the top, middle, and bottom payoffs are to player \(1,2,3\), respectively. Find all subgame-perfect equilibria (SPE) in mixed strategies. 1 2 a R 3 2 b A B A B 202 231 333 4 1 1 +3 122
Consider this three-player game in extensive form. Find all subgame-perfect equilibria (SPE) in mixed strategies. 2 1 R A 3 3 B 3 D 202 231 333 4 1 3 143 121
Consider a bargaining set \(S\) with threat point \((0,0)\). Let \(a>0, b>0\), and let \(S^{\prime}=\{(a u, b v) \mid(u, v) \in S\}\). The set \(S^{\prime}\) is the set of utility pairs in \(S\) re-scaled by \(a\) for player I and by \(b\) for player II. Recall that one axiomatic property of the
Consider the following two bimatrix games (i) and (ii):For each bimatrix game, do the following:(a) Find max-min strategies \(\hat{x}\) and \(\hat{y}\) (which may be mixed strategies) and corresponding max-min values \(u_{0}\) and \(v_{0}\) for player I and II, respectively.(b) Draw the convex hull
We repeat the definition of the discretized ultimatum game. Let \(M\) be a positive integer. Player I's possible actions are to ask for a number \(x\), called his demand, which is one of the (integer) numbers \(0,1,2, \ldots, M\). In response to this demand, player II can either accept (A) or
Consider the following bargaining problem. In the usual way, a "unit pie" is split into non-negative amounts \(x\) and \(y\) with \(x+y \leq 1\). The utility function of player \(\mathrm{I}\) is \(u(x)=x\), the utility function of player II is \(v(y)=1-(1-y)^{2}\). The threat point is \((0,0)\).(a)
Consider the bargaining problem of splitting a unit pie with utility functions \(u, v:[0,1] \rightarrow[0,1]\) as in Exercise 11.4, that is, \(u(x)=x\) and \(v(y)=\) \(1-(1-y)^{2}\). Assume now that the bargaining behavior is described by the subgameperfect equilibrium of the standard
Find the Nash and correlated equilibria of the following zero-sum game (with payoffs to the row player): I II 4 T 2 B 0 3
Consider the following variant of the Rock-Paper-Scissors game where two equal strategies tie with payoff 0 for both players, but two unequal strategies give payoff 1 to the winner and -2 to the loser.(a) Find a Nash equilibrium of this game, and its payoff to the two players. Show that when this
For the game in Figure 12.4, the coarse correlated equilibrium \(z\) on the right has only two probabilities \(a=z_{32}\) for the strategy pair \((B, M)\) and \(b=z_{23}\) for the strategy pair \((M, B)\) that are positive; all other components \(z_{i j}\) are zero. Find all possible values of
Show that in an \(N\)-player game where every player has two strategies, every coarse correlated equilibrium is a correlated equilibrium.
Show that if \(Y\) is a Markov chain on two states, then the off-diagonal elements of \(Y\), in their columns, are proportional to the probabilities of a stationary distribution \(x\) of \(Y\). That is, ifthen \(\left(\begin{array}{ll}c & b\end{array}\right)=(c+b) x^{\top}\). Exactly when is
Let \(x \in \mathbb{R}^{m}\) and \(y \in \mathbb{R}^{n}\) and remember that all vectors are column vectors. If \(m=n\), what is the difference between \(x^{\top} y\) and \(x y^{\top}\) ? What if \(m eq n\) ?
Find all mixed equilibria (which always includes any pure equilibria) of this \(3 \times 2\) game. I T M B II 0 2 3 1 1 0 3 9 5 3 r 0 2 4
Consider the three-player game tree in Exercise 4.5. Recall that some or all of the strategies in a mixed equilibrium may be pure strategies.(a) Is the following statement true or false? It is easiest to argue directly with the game tree.For each of the players I, II, or III, the game has a mixed
In this \(2 \times 2\) game, \(A, B, C, D\) are the payoffs to player I, which are real numbers, no two of which are equal. Similarly, \(a, b, c, d\) are the payoffs to player II, which are real numbers, also no two of which are equal.(a) Under which conditions does this game have a mixed
Consider the following \(2 \times 5\) game:(a) Draw the expected payoffs to player II for all her strategies \(a ,b, c, d, e\), in terms of the probability \(p\), say, that player I plays strategy \(B\). Indicate the best responses of player II, depending on that probability \(p\).(b) Using the
Consider the Quality game in Figure 3.2(b).Figure 3.2(b)(a) Use the "goalpost" method twice, to draw the upper envelope of best-response payoffs to player I against the mixed strategy of player II, and vice versa.(b) Explain which of these diagrams shows that a pure strategy dominates another pure
Find all equilibria of this degenerate game, which is a variant of the Inspection game in Figure 6.2. The difference is that player II, the inspectee, derives no gain from acting illegally (playing \(r\) ) even if she is not inspected (player I choosing \(T\) ). I T II 1 r 0 0 0 B -1 -10 0 -6 -90
Prove Proposition 6.8.
Let \(C=\left\{(x, y) \in \mathbb{R}^{2} \mid x^{2}+y^{2} \leq 1\right\}\), and consider the following functions \(g, h, f: C \rightarrow C:\)Let \(G=g(C)=\{g(x, y) \mid(x, y) \in C\}\). Describe the geometric shape of \(C\) and \(G\), and how the functions \(g\) and \(h\) act. Does each of the
For a general polytope \(P\), a vertex of \(P\) is defined as an extreme point of \(P\), that is, it is not a convex combination of other points of \(P\).(a) Show that for a simplex \(S\) given as the convex hull of the affinely independent points \(v^{1}, \ldots, v^{d}\), each point \(v^{i}\) is
Give examples of(a) two simplices whose intersection is not a simplex;(b) a nonempty convex set that has no extreme points;(c) a bounded convex set that has at least one extreme point but that is not equal to the convex hull of its extreme points.
Consider the following Penalty game. It gives the probability of scoring a goal when the row player (the striker) adopts one of the strategies \(L\) (shoot left), \(R\) (shoot right), and the column player (the goalkeeper) uses the strategies \(l\) (jump left), \(w\) (wait then jump), \(r\) (jump
(a) Consider a \(2 \times n\) zero-sum game where the best response to every pure strategy is unique. Suppose that the top row and leftmost column define a pure-strategyequilibrium of the game. Show that this equilibrium can be found by iterated elimination of strictly dominated strategies.(b) Does
Consider the following simultaneous game between two players: Each player chooses an integer between 1 and 5 (e.g., some number of fingers shown with your hand), and the higher number wins, except when it is just one larger than the lower number, in which case the lower number wins. So 4 beats 2
Consider the following bimatrix game.(a) Find an equilibrium of this game in mixed (including pure) strategies.(b) Find a max-min strategy of player I, which may be a mixed strategy, and the resulting max-min payoff to player I.(c) Find a max-min strategy of player II, which may be a mixed
Consider the following congestion network with origin and destination nodes \(o\) and \(d\), and an intermediate node \(v\). There are ten individual users who want to travel from \(o\) to \(d\).As shown, the cost on the bottom edge is \(c(x)=x+2\) when it has \(x\) users, where \(x+y=10\). The
Consider the following two congestion networks with the indicated cost functions for a flow \(x\) on the respective edge; for example, \(50+x\) is short for \(c(x)=50+x\). Note that different edges may have different flows \(x\) that depend on the chosen routes by the users.Suppose there are six
Consider the following congestion network with three nodes. As shown, an edge has the cost function \(c(x)=x\) if the flow (number of users) on that edge is \(x\), except for the edges \(v u\) and \(w u\) where \(c(x)=0\). The table on the right shows four users \(i=1,2,3,4\) with different origins
Consider the following \(3 \times 3\) game.(a) Identify all pairs of strategies where one strategy weakly dominates the other.(b) Assume you are allowed to remove a weakly dominated strategy of some player. Do so, and repeat this process (of iterated elimination of weakly dominated strategies)
Consider the following three-player game in strategic form.Each player has two strategies: Player I has the strategies \(T\) and \(B\) (the top and bottom row), player II has the strategies \(l\) and \(r\) (the left and right column in each \(2 \times 2\) panel), and player III has the strategies
Consider the duopoly game as studied in Section 3.6 where player I and II produce nonnegative quantities \(x\) and \(y\) and the unit price on the market is \(12-x-y\). In addition, the players now have a cost of production, which is a cost of 1 per unit for player I and a cost of 2 per unit for
Each of \(N\) people chooses whether or not to contribute a fixed amount towards the provision of a public good. The good is provided if and only if at least \(K\) people contribute, where \(2 \leq K \leq N\). If the good is not provided, contributions are not refunded. Each person strictly ranks
Consider the following game tree. As always, the top payoffs at a leaf are for player I and bottom payoffs for player II.(a) What is the number of strategies of player I and of player II?(b) How many reduced strategies do they have?(c) Give the reduced strategic form of the game.(d) What are the
Consider the following game tree.(a) What is the number of strategies of player I and of player II? How many reduced strategies does each of the players have?(b) Give the reduced strategic form of the game.(c) What are the equilibria of the game in reduced strategies? What are the subgameperfect
Consider the following game trees.(a) Find all equilibria for the game tree on the left. Which of these are subgameperfect?(b) In the game tree on the right, the payoffs \(a,b, c, d\) are positive real numbers. For each of the following statements (i), (ii), (iii), decide if it is true or false,
Consider the following two game trees (a) and (b). Payoffs have been omitted because they are not relevant for the question. In each case, how many strategies does each player have? How many reduced strategies? (a) II A B/CD E II II II II C 8 k (b) a II Y II d I Q R ST
Consider the following three-player game tree. At a leaf, the topmost payoff is to player I, the middle payoff is to player II, and the bottom payoff is to player III.(a) How many strategy profiles does this game have?(b) Identify all pairs of strategies where one strategy strictly, or weakly,
Consider a game \(G\) in strategic form. Recall that the commitment game derived from \(G\) is defined by letting player I choose one of his strategies \(x\), which is then announced to player II, who can then in each case choose one of her strategies in \(G\) as a response to \(x\). The resulting
Let \(G\) be the following game: Player I chooses a non-negative real number \(x\), and simultaneously player II chooses a non-negative real number \(y\). The resulting (symmetric) payoffs are(a) Given \(x\), determine player II's best response \(y(x)\) (which is a function of \(x\) ), and player
Find positive weights \(w\) in a utility function of the form \(u\left(c_{1}, c_{2}\right)=\) \(c_{1}+w \cdot c_{2}\) to create at least four different strict rankings among the candidates \(x, y, z\) in (5.10). [Hint: Experiment with some values of \(w\) and vary them. What are the values of
Complete the proof of Proposition 5.2 for the third case \(x_{1}
(a) Prove the equivalence (5.25), including the inequalities, for real numbers \(A, B, C\) that fulfill \(A
For this exercise, be careful to use only the definitions and not your intuitions that are familiar to you about the symbols \(\leq\) and \(
Consider the game Nim with heaps of tokens. The players alternately remove some tokens from one of the heaps. The player to remove the last token wins. Try to prove part (a) without reference to the binary representation for playing Nim optimally.(a) For all positions with three heaps, where one of
Misère Nim is played just like Nim but where the last player to move loses. A losing position is therefore a single Nim heap with a single token in it (which the player then has to take). Another losing position is given by three heaps with a single token each.(a) Determine the winning and losing
The impartial game Cram is played on a board of \(m \times n\) squares, where players alternately place a domino on the board which covers two adjacent squares that are free (not yet occupied by a domino), vertically or horizontally. The first player who cannot place a domino any more loses.
Consider the following game Chomp: A rectangular array of \(m \times n\) dots is given, in \(m\) rows and \(n\) columns, like \(3 \times 4\) in the next picture on the left. A dot in row \(i\) (from the top) and column \(j\) (from left) is named \((i, j)\). A move consists in picking a dot \((i,
(a) Complete the entries of equivalent Nim heaps for the Queen-move game in columns 5 and 6, rows 0 to 3, in the table in Figure 1.4 (assuming the queen can be anywhere on the board).(b) Describe all winning moves in the game sum of the Queen-move game and the Nim heap in Figure 1.3.Figure 1.3 3 2
Consider the game Cram from Exercise 1.4, played on a \(1 \times n\) board for \(n \geq 2\). Let \(D_{n}\) be the Nim value of that game, so that the starting position of the \(1 \times n\) board is equivalent to a Nim heap of size \(D_{n}\). For example, \(D_{2}=1\) because the \(1 \times 2\)
Consider the following game on a rectangular board where a white and a black counter are placed in each row, like in this example:Player I is White and starts, and player II is Black. Players take turns. In a move, a player moves a counter of his color to any other square within its row, but may
Consider the following network (in technical terms, a directed graph or "digraph"). Each circle, here marked with one of the letters A to P, represents a node of the network. Some of these nodes (here A, F, G, H, and K) have counters on them. There can be any number of counters on a node, like
Consider the game Chomp from Exercise 1.5 of size \(2 \times 4\), in a game sum with a Nim heap of size 4.What are the winning moves of the starting player I, if any? 00
In \(m \times n\) Cram (see Exercise 1.4), a rectangular board of \(m \times n\) squares is given. The two players alternately place a domino either horizontally or vertically on two unoccupied adjacent squares, which then become occupied. The last player to be able to place a domino wins.(a) Find
Consider the following variant of Nim called Split-Nim, which is played with heaps of tokens as in Nim. Like in Nim, a player can remove some tokens from one of the heaps, or else split a heap into two (not necessarily equal) new heaps. For example, a heap of size 4 can be reduced to size 3, 2, 1,
The impartial game Gray Hackenbush is played on a figure consisting of nodes and edges that are connected to these nodes or to the ground (the ground is the dashed line in the pictures below). A move is to remove an edge, and with it all the edges that are then no longer connected to the ground.
This exercise is about partizan games as treated in Section 1.8.(a) Complete the proof of (1.32) by showing that Right as starting player loses.The partizan game (Black-White) Hackenbush is played on a figure consisting of nodes and black or white edges that are connected to these nodes or to the
(a) Show that if \(G\) and \(H\) are partizan games and Left wins when moving second in \(G\) and in \(H\), then Left wins when moving second in \(G+H\).(b) Give an example of games \(G\) and \(H\) so that Left wins when moving first in \(G\) and in \(H\), but cannot win in \(G+H\), even if Left

Game Theory Basics 1st Edition Bernhard Von Stengel - Solutions

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