Questions and Answers of Basic Mathematics

(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
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
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,
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
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
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
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
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
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
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
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 \(S\)
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
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
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
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
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
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,
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
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
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
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
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
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,
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
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
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
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
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
(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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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,
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)
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,
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
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
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
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
(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
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
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
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
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
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
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
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) 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 \(H\)