Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
values |
|---|---|---|
Theorem 1. Suppose that there are value functions \( {W}_{j}^{ * }\left( {t,\mathbf{x}}\right) \) and feedback strategies \( \left( {{\sigma }_{j}^{ * },{\sigma }_{-j}^{ * }}\right) \) which satisfy the local-game equilibrium conditions defined in equations (4.39),(4.43) for \( t = 0,1,2,\ldots, T - 1,\mathbf{x} \in {\... | For a proof see Haurie et al. (2012). | No |
Lemma 5. The current-valued value functions \( {V}_{j}^{ * }\left( \mathbf{x}\right) \) and \( {V}_{-j}^{ * }\left( \mathbf{x}\right) \), defined in (4.51), satisfy the following recurrence equations, backward in time, also known as Bellman equations:\n\n\[ \n{V}_{j}^{ * }\left( {\mathbf{x}{\left( \mathbf{t}\right) }^{... | The proof is identical to that proposed for Lemma 4 (see Haurie et al. 2012), except for obvious changes in the definition of current-valued value functions. | Yes |
Theorem 2. Consider value functions \( {V}_{j}^{ * }\left( \mathbf{x}\right) \) and \( {V}_{-j}^{ * }\left( \mathbf{x}\right) \) and a stationary feedback strategy vector \( \left( {{\sigma }_{j}^{ * },{\sigma }_{-j}^{ * }}\right) \), such that the following holds true:\n\n\[ \n{V}_{j}^{ * }\left( \mathbf{x}\right) = \... | For a proof see Haurie et al. (2012). | Yes |
Lemma 6. Feedback-equilibrium strategies (4.68) lead to the fishery's extinction when the steady state of (4.71) is asymptotically stable, i.e., when\n\n\[ a < \frac{2}{\beta } - 1 \]\n | Proof. We need to show that\n\n\[ \lambda = a\frac{1 - \sqrt{1 - a{\beta }^{2}}}{1 + \sqrt{1 - a{\beta }^{2}}} < 1 \]\n\n(4.72)\n\nThis inequality implies that\n\n\[ a\left( {1 - \sqrt{1 - a{\beta }^{2}}}\right) < 1 + \sqrt{1 - a{\beta }^{2}} \]\n\n\[ a - 1 < \left( {a + 1}\right) \sqrt{1 - a{\beta }^{2}} \]\n\n\[ \lef... | Yes |
Lemma 7. Feedback-equilibrium strategies (4.68) do not lead to the fishery's extinction if\n\n\[ \n\\frac{2}{\\beta } - 1 < a < \\frac{1}{{\\beta }^{2}}\n\]\n\nWhen the above condition is satisfied, the steady state of (4.71) is unstable. | Proof. We require\n\n\[ \n\\lambda = a\\frac{1 - \\sqrt{1 - a{\\beta }^{2}}}{1 + \\sqrt{1 - a{\\beta }^{2}}} > 1.\n\]\n\n(4.75)\n\nThe previous lemma established that the (zero) unique steady state is unstable if\n\n\[ \na > \\frac{2}{\\beta } - 1\n\]\n\nOn the other hand, (4.67) has to be satisfied in order to ensure ... | Yes |
When Player 1 is the leader, what is the solution to the game under open-loop information? | When Player 1 is the leader, the solution to the game is entry \\{ 4,1\\} (i.e., fourth row, first column) in Table 4.1. The corresponding costs to the leader and follower are 6 and 5 , respectively. This solution defines the following control and state sequence:\n\n\\[ \n\\left. \\begin{array}{l} {u}_{1}\\left( 0\\rig... | Yes |
Compute the Stackelberg feedback equilibrium for the game shown in Fig. 4.3. | At time \( t = 1 \), there are three possible states. From every state, the transition to the next stage \( \left( {t = 2}\right) \) is a \( 2 \times 2 \) bimatrix game. In fact, Table 4.2 is one of them: it determines the last-stage Stackelberg equilibrium from state \( x = 0 \) . The policies and the costs to the pla... | Yes |
Lemma 8. A pair of linear feedback-Nash equilibrium strategies (4.111) and a pair of value functions (4.110) which satisfy equations (4.108) and (4.109) uniquely exist if\n\n\[ \n{a}^{v} < \frac{1}{\beta } < a\;\text{ and }\;v < \frac{1}{2}.\n\]\n\nMoreover, if \( B/D \) is sufficiently large, the growth rate \( \mu = ... | For a proof see Krawczyk and Shimomura (2003) or Haurie et al. (2012). | No |
Lemma 9. A pair of linear Pareto-optimal strategies uniquely exists if and only if (4.132) is satisfied. Moreover, the growth rate \( \mu \) corresponding to the strategy pair is positive and time invariant, even if \( B = 0 \) . | For a proof see Krawczyk and Shimomura (2003) or Haurie et al. (2012). | No |
Proposition 2. Under the above assumptions the value function \( {v}^{ * } \) is upper semianalytic. Player 2 has a universally measurable optimal strategy and, for any \( \varepsilon > 0 \), player 1 has a universally measurable \( \varepsilon \) -optimal strategy. If, in addition, we assume that \( A\left( x\right) \... | As a corollary to Theorem 5.1 in Nowak (1986), we can state the following result. | No |
Proposition 3. Assume that \( x \rightarrow A\left( x\right) \) is lower semicontinuous and has complete values in \( A \) and \( x \rightarrow B\left( x\right) \) is upper semicontinuous and compact valued. If \( r \) : \( K \rightarrow \mathbb{R} \) is lower semicontinuous on \( K \), then \( {v}^{ * } \) is lower se... | The lower semicontinuity of \( {v}^{ * } \) in Proposition 3 is a corollary to the maximum theorem of Berge (1963). | No |
Theorem 1. Assume (C1)-(C3),(W1)-(W2), and (M1)-(M2). Then \( {v}_{\beta } \in {\underline{U}}_{\omega }\left( X\right) \) , \( {T}_{\beta }{v}_{\beta } = {v}_{\beta } \) and there exists a stationary strategy \( {f}^{ * } \in F \) such that\n\n\[ \n{v}_{\beta }\left( x\right) = \mathop{\inf }\limits_{{b \in B\left( {x... | The proof of Theorem 1 consists of two steps. First, we deal with truncated models, in which the payoff function \( u \) is replaced by \( {u}_{k} \) . Then, making use of the fixed point argument, we obtain an upper semicontinuous solution to the Bellman equation, say \( {v}_{\beta, k} \) . Next, we observe that the s... | No |
Consider the maxmin control model, where \( X = \lbrack 0,\infty ), A\left( x\right) = \left\lbrack {0, x}\right\rbrack \) , \( B\left( {x, a}\right) = B \), and \( u\left( {x, a, b}\right) = {a}^{\sigma } \) for \( \left( {x, a, b}\right) \in K \) . Then, the transition probability \( q \) is of the form\n\n\[ q\left(... | Define now\n\n\[ \omega \left( x\right) = {\left( r + x\right) }^{\sigma },\;x \in X, \] \n\n(5.9)\n\nwhere \( r \geq 1 \) is a constant. Clearly, \( {u}^{ + }\left( {x, a, b}\right) = {a}^{\sigma } \leq \omega \left( x\right) \) for any \( \left( {x, a, b}\right) \in K \) . Hence, condition (M2) is satisfied. Moreover... | Yes |
Let us consider again the model from Example 1 but with \( u\left( {x, a, b}\right) = \) \( \ln a, a \in A\left( x\right) = \left\lbrack {0, x}\right\rbrack \) . This utility function has a number of applications in economics (see Stokey et al. 1989). Nonetheless, the two-sided weighted norm approach cannot be employed... | Clearly, \( {u}^{ + }\left( {x, a, b}\right) = \max \{ 0,\ln a\} \leq \max \{ 0,\ln x\} \leq \omega \left( x\right) \) for all \( \left( {x, a, b}\right) \in K \) . By Jensen’s inequality it follows that\n\n\[ \n{\int }_{X}\omega \left( y\right) q\left( {{dy} \mid x, a, b}\right) = {\int }_{0}^{\infty }\omega \left( {\... | Yes |
Consider the model from Example 1 with the following state evolution equation:\n\n\[ \n{x}_{n + 1} = \left( {1 + {\rho }_{0}}\right) \left( {{x}_{n} - {a}_{n}}\right) + {\xi }_{n},\;n \in \mathbb{N}, \n\]\n\nwhere \( {\rho }_{0} \) is constant introduced in Example 2. The transition probability \( q \) is now of the fo... | Let the function \( \omega \) be as in (5.9). Applying Jensen’s inequality we obtain\n\n\[ \n{\int }_{X}\omega \left( y\right) q\left( {{dy} \mid x, a, b}\right) = {\int }_{0}^{\infty }\omega \left( {\left( {x - a}\right) \left( {1 + {\rho }_{0}}\right) + s}\right) b\left( {ds}\right) \n\]\n\n\[ \n\leq \omega \left( {x... | Yes |
Theorem 4. Assume (C1), (W1)-(W2), and (M1)-(M2). In addition, let the correspondence \( x \rightarrow B\left( x\right) \) be lower semicontinuous and let every set \( B\left( x\right) \) be a complete subset of \( B \) . Then, the game has a value \( {v}_{\beta } \in {\underline{U}}_{\omega }\left( X\right) \), player... | The assumption that every \( B\left( x\right) \) is complete in \( B \) is made to assure that \( G \neq \) \( \varnothing \) (see Kuratowski and Ryll-Nardzewski 1965). The construction of an \( \varepsilon \) -optimal semistationary strategy for player 2 is based on using \ | No |
Theorem 7. Assume (5.21), (C8)-(C9), and (W3). Then, the positive stochastic game has a value function \( {v}_{p} \) which is lower semicontinuous and \( {v}_{p}\left( x\right) = \) \( \mathop{\sup }\limits_{{\beta \in \left( {0,1}\right) }}{v}_{\beta }\left( x\right) \) for all \( x \in X \) . Moreover, \( {v}_{p} \) ... | The proof of Theorem 7 is similar to that of Theorem 6 and makes use of Proposition 3. | No |
Example 4. Let \( X = \{ - 1,0,1\}, A = \{ 0,1\}, B = \{ 0,1\} \) . States \( x = - 1 \) and \( x = 1 \) are absorbing with zero payoffs. If \( x = 0 \) and both players choose the same actions \( \left( {a = 1 = b\text{or}a = 0 = b}\right) \), then \( u\left( {x, a, b}\right) = 1 \) and \( q\left( {-1 \mid 0, a, b}\ri... | In state \( x = 0 \) we obtain the equation \( {v}_{p}\left( 0\right) = 1/\left( {2 - {v}_{p}\left( 0\right) }\right) \) , which yields the solution \( {v}_{p}\left( 0\right) = 1 \) . In this game player 1 has no optimal strategy. | Yes |
Example 5. Let \( X = \mathbb{N} \cup \{ 0\}, A = \{ 1,2\}, B = \{ 1,2\} \) . State \( x = 0 \) is absorbing with zero payoffs. Let \( x \geq 2 \) and \( a = 1 \) . Then \( u\left( {x,1, b}\right) = 0 \) for \( b \in B \) and \( q\left( {x - 1 \mid x,1,1}\right) = q\left( {x + 1 \mid x,1,2}\right) = 1 \) . If \( x \geq... | It is shown that \( {v}_{p}\left( x\right) = \left( {x + 1}\right) /{2x} \) for \( x \geq 2 \) and player 1 has no stationary \( \varepsilon \) -optimal strategy. It is easy to check that the function \( {v}_{p} \) given here is a solution to equation (5.22). It may be interesting to note that also \( v\left( 0\right) ... | No |
Theorem 8. Assume (C10)-(C13), (GE1)-(GE3), and (W2). Then, the following hold:\n\n(a) There exist a constant \( v \) and \( {h}^{ * } \in {\mathcal{B}}_{W}\left( X\right) \), which is continuous and such that\n\n\[ \n{h}^{ * }\left( x\right) = \operatorname{val}\left\lbrack {u\left( {x,\cdot , \cdot }\right) - {v\tau ... | The proof of Theorem 8 owes much to the approach introduced by Vega-Amaya (2003), who used a fixed point argument in the game model with setwise continuous transition probabilities. However, we cannot directly apply a fixed point argument. First, we have to regularize (to smooth in some sense) certain functions. Using ... | Yes |
Theorem 9. Assume (C10)-(C13), (GE1)-(GE3), (W2), and (R1)-(R2). Then, \( v \) is the value of the game and the pair of stationary strategies \( \left( {\widehat{f},\widehat{g}}\right) \) is also optimal for the players in the game with the time average payoff defined in (5.25). | The proof of Theorem 9 requires different methods than the proof of Theorem 8 and was formulated as Theorem 5.1 in Jaskiewicz (2009). The point of departure of its proof is the optimality equation (5.28). It allows to define a certain martingale or a super- (sub-) martingale, to which the optional sampling theorem is a... | No |
Theorem 10. Assume (C10)-(C15), (W2), and (GE1)-(GE2). Then, the pair of optimal strategies \( \left( {\bar{f},\bar{g}}\right) \in F \times G \) from Theorem 8 is sample path optimal with respect to each of the payoffs in (5.29),(5.30), and (5.31). Moreover, \( {\widehat{J}}^{1}\left( {x,\bar{f},\bar{g}}\right) = \) \(... | The point of departure in the proof of Theorem 10 is the optimality equation from Theorem 8. Namely, from (5.28) we get two inequalities. The first one is obtained with the optimal stationary strategy \( \bar{f} \) for player 1, whereas the second one is connected with the optimal stationary strategy \( \bar{g} \) for ... | No |
Example 7. Let \( X = \{ 1,2\}, A\left( x\right) = B\left( x\right) = \{ 1,2\} \) for \( x \in X \) . Assume that state \( x = 2 \) is absorbing with zero payoffs. In state \( x = 1 \), we have \( u\left( {1,1,1}\right) = 2 \) , \( u\left( {1,2,2}\right) = 6 \), and \( u\left( {1, i, j}\right) = 0 \) for \( i \neq j \)... | \[ {w}_{\lambda }\left( 1\right) = \operatorname{val}\left\lbrack \begin{array}{ll} 1 + \frac{1}{2}{w}_{\lambda }\left( 1\right) & 0 + \frac{1}{2}{w}_{\lambda }\left( 2\right) \\ 0 + \frac{1}{2}{w}_{\lambda }\left( 2\right) & 3 + \frac{1}{2}{w}_{\lambda }\left( 1\right) \end{array}\right\rbrack . \] Clearly, \( {w}_{\l... | Yes |
Theorem 18. If \( \left( {{\phi }_{1}^{ * },{\phi }_{2}^{ * },{v}_{1}^{ * },{v}_{2}^{ * },{f}^{ * },{g}^{ * }}\right) \) is a feasible solution of (NP2) with the property that \( \mathop{\sum }\limits_{{x \in X}}\left( {{v}_{1}\left( x\right) + {v}_{2}\left( x\right) }\right) = 0 \), then it is a global minimum and \( ... | For a proof consult Filar et al. (1991) or pages 127-129 in Filar and Vrieze (1997). | No |
Theorem 21. Let \( {\mathcal{G}}_{0} \) be a recursive game such that player 1 is more informed than player 2. Then, for every initial distribution \( p \in \Pr \left( {X \times S \times T}\right) \), both the asymptotic value and the uniform maxmin exist and are equal, i.e., | \[ {\underline{v}}_{\infty } = \mathop{\lim }\limits_{{n \rightarrow \infty }}{v}_{n} = \mathop{\lim }\limits_{{lambda \rightarrow 0}}{v}_{\lambda } \] | No |
Theorem 22. If \( C \subset {\mathbb{R}}^{k} \) is a nonempty closed set satisfying the Blackwell condition, then \( C \) is approachable in game \( {G}_{\infty } \) . | An approachability strategy is \( \widehat{\pi }\left( {h}_{n}\right) = {s}_{1}\left( {\bar{g}}_{n}\right) \), where \( {h}_{n} \) is the history of a play at stage \( n \) . | No |
Proposition 2. Let \( b : X \rightarrow {\mathbb{R}}^{n} \) be a Borel measurable selection of the mapping \( x \rightarrow \operatorname{co}\mathcal{N}\mathcal{P}\left( x\right) \) . Then, there exist Borel measurable selections \( {b}^{i} : X \rightarrow {\mathbb{R}}^{n} \) and Borel measurable functions \( {\lambda ... | Similarly as in Nowak and Raghavan (1992), from Filippov's measurable implicit function theorem, we conclude the following facts. | No |
Proposition 4. If \( b : X \rightarrow {\mathbb{R}}^{n} \) is a Borel measurable selection of the mapping \( x \rightarrow \operatorname{co}\mathcal{N}\mathcal{P}\left( x\right) \), then there exist Borel measurable selections \( {\psi }^{i} \) of the mapping \( x \rightarrow \mathcal{N}\left( x\right) \) and Borel mea... | \[ b\left( x\right) = \mathop{\sum }\limits_{{i = 1}}^{{n + 1}}{\lambda }^{i}\left( x\right) \left( {{P}^{1}\left( {x,{\psi }^{i}\left( x\right) }\right) ,\ldots ,{P}^{n}\left( {x,{\psi }^{i}\left( x\right) }\right) }\right) . \] | No |
Proposition 5. Let \( \mu \) be a nonatomic Borel probability measure on \( X \) . Assume that \( {q}_{j}\left( {j = 1,\ldots, l}\right) \) are Borel measurable transition probabilities from \( X \) to \( X \) and for every \( j \) and \( x \in X,{q}_{j}\left( {\cdot \mid x}\right) \ll \mu \), i.e., \( {q}_{j}\left( {\... | Moreover, there exists a Borel measurable mapping \( \phi : X \times X \rightarrow \Pr \left( A\right) \) such that \( \phi \left( {x, y}\right) \in \mathcal{N}\left( x\right) \) for all \( x, y \in X \) . | Yes |
Theorem 1. Every finite-stage nonzero-sum discounted stochastic game satisfying the above conditions has a subgame-perfect equilibrium. For any \( \varepsilon > 0 \), there exists an \( \varepsilon \) -equilibrium \( {\pi }^{\varepsilon } \) in Markov strategies, i.e., | \[ {J}_{\beta }^{i}\left( {x,{\pi }^{\varepsilon }}\right) + \varepsilon \geq {J}_{\beta }^{i}\left( {x,\left( {{\pi }_{i},{\pi }_{-i}^{\varepsilon }}\right) }\right) \;\text{ for all }\;x \in X,{\pi }_{i} \in {\Pi }_{i}\text{ and }i \in N. \] | Yes |
Theorem 2. Every discounted nonzero-sum stochastic game \( G \) satisfying (A1) has a subgame-perfect equilibrium. | Theorem 2 was proved in a more general form by Mertens and Parthasarathy (2003), where the payoffs and discount factors may depend on time and the state space is a general measurable space. A special case was considered by Mertens and Parthasarathy (1991), who assumed that the action sets are finite and state independe... | No |
Theorem 3. Let the payoff functions \( {g}_{i}, i \in N \) be bounded, Borel measurable, and DS-continuous on \( {H}_{\infty } \) and let the action spaces \( {A}_{i}, i \in N \) be finite. Then the game has a subgame-perfect equilibrium. | The proof of Theorem 3 applies some techniques from gambling theory described in Dubins and Savage (2014), i.e., approximations of \( {DS} \) -continuous functions by \ | No |
Theorem 5. Any discounted stochastic game \( G \) satisfying (A2) has a stationary correlated equilibrium with public signals. | Theorem 5 was proved by Nowak and Raghavan (1992). First it is shown by making use of a theorem in Glicksberg (1952) that the correspondence \( v \rightarrow {\mathcal{M}}_{v} \) has a fixed point, i.e., there exists \( {w}^{ * } \in {B}^{n}\left( X\right) \) such that \( {w}^{ * }\left( x\right) \in \operatorname{co}{... | Yes |
Theorem 6. Assume that game \( G \) satisfies (A3). Then, \( G \) has a stationary almost Markov perfect equilibrium. | We outline the proof of Theorem 6 for nonatomic measure \( \mu \) . The general case needs an additional notation. First, we show that there exists a Borel measurable mapping \( {w}^{ * } \in {B}^{n}\left( X\right) \) such that \( {w}^{ * }\left( x\right) \in \operatorname{co}{\mathcal{{NP}}}_{{w}^{ * }}\left( x\right)... | No |
Consider a game where the set \( A \) is finite and the transition probability \( q \) is Borel measurable. Then, the game has a stationary almost Markov perfect equilibrium. | We show that the game meets (A3). Let \( m \in \mathbb{N} \) be such that \( A = \) \( \left\{ {{a}^{1},\ldots ,{a}^{m}}\right\} \) . Now, for \( j = 1,\ldots, m \), define\n\n\[ \n{\alpha }_{j}\left( {s, a}\right) \mathrel{\text{:=}} \left\{ {\begin{array}{l} 1,\text{ if }a \in A\left( x\right), a = {a}^{j} \\ 0,\text... | Yes |
Theorem 11. Every discounted stochastic game satisfying assumptions (S1)-(S3) has a pure stationary Markov perfect equilibrium. | The proof of Theorem 11 uses the fact that the auxiliary game \( {\Gamma }_{v}\left( x\right) \) has a unique Nash equilibrium for any vector \( v = \left( {{v}_{1},\ldots ,{v}_{n}}\right) \) of nonnegative continuation payoffs \( {v}_{i} \) such that \( {v}_{i}\left( 0\right) = 0 \) . The uniqueness follows from page ... | Yes |
Example 1 (Dynamic Cournot oligopoly). Let \( X = \left\lbrack {0,\bar{x}}\right\rbrack \) and \( x \in X \) represent a realization of a random demand shock that is modified at each period of the game. Player \( i \in N \) (oligopolist) sets a production quantity \( {a}_{i} \in {A}_{i}\left( x\right) = \left\lbrack {0... | However, the set of Nash equilibria in \( {\Gamma }_{v}\left( x\right) \) may contain many points. A modification of the proof of Theorem 1 given in Jaskiewicz and Nowak (2016) implies that this game has a pure stationary almost Markov perfect equilibrium. | Yes |
Example 2 (Cournot competition with substituting goods in differentiated markets). This model is inspired by a dynamic game with complementary goods studied by Curtat (1996). Related static games were already discussed in Spence (1976) and Vives (1990). There are \( n \) firms on the market and firm \( i \in N \) produ... | \[ {u}_{i}\left( {x, a}\right) \mathrel{\text{:=}} {a}_{i}\left\lbrack {{P}_{i}\left( a\right) - {c}_{i}\left( {x}_{i}\right) }\right\rbrack ,\;a = \left( {{a}_{1},\ldots ,{a}_{n}}\right) ,\;x = \left( {{x}_{1},\ldots ,{x}_{n}}\right) \in X. \] The transition probability of the next state (experience vector) is of the ... | Yes |
Example 3 (Cournot oligopoly with complementary goods in differentiated markets). Let \( h \) be defined as in (6.5). Suppose that the payoff function in the game \( {\Gamma }_{v}\left( x\right) \) satisfies the following condition:\n\n\[ \frac{{\partial }^{2}{U}_{\beta }^{i}\left( {{v}_{i}, x, a}\right) }{\partial {a}... | Then, by Theorem 4 in Milgrom and Roberts (1990), the game \( {\Gamma }_{v}\left( x\right) \) is supermodu-lar. Note that within our framework, it is sufficient to prove that for \( {u}_{i}\left( {x, a}\right) \), defined in (6.3), it holds \( \frac{{\partial }^{2}{u}_{i}\left( {s, a}\right) }{\partial {a}_{i}\partial ... | Yes |
Proposition 7. Other condition imposed by Curtat (1996) states that the payoff functions \( {u}_{i}\left( {x, a}\right) \) are increasing in \( {a}_{-i} \) and, more importantly, satisfy the so-called strict diagonal dominance condition for each \( x \in X \) . For details the reader is referred to Curtat (1996) and Ro... | The result in Curtat (1996) on the existence of stationary Markov perfect equilibria for supermodular discounted stochastic games is based upon the lattice theoretic arguments and on complementarity and monotonicity assumptions. The state and action spaces are assumed to be compact intervals in Euclidean space, and the... | No |
Example 4. Let \( X = \left\lbrack {0,1}\right\rbrack ,{A}_{i}\left( x\right) = \left\lbrack {0,1}\right\rbrack \) for all \( x \in X \) and \( i \in N = \{ 1,2\} \) . We consider the symmetric game where the stage utility of player \( i \) is\n\n\[ \n{u}_{i}\left( {x,{a}_{1},{a}_{2}}\right) = {a}_{1} + {a}_{2} + {2x}{... | We assume that \( {\mu }_{1} \) has the density \( {\rho }_{1}\left( y\right) = {2y} \) and \( {\mu }_{2} \) has the density \( {\rho }_{2}\left( y\right) = \) \( 2 - {2y}, y \in X \) . Note that \( {\mu }_{1} \) stochastically dominates \( {\mu }_{2} \) . From the definition of \( q \), it follows that higher states \... | Yes |
Theorem 12. If conditions (A4)-(A6) are satisfied, then the limit-average payoff \( n \) -person stochastic game has a stationary Markov perfect equilibrium. | The above result follows from Theorem 2.6 in Altman et al. (1997), where it is also shown that under (A4)-(A6) any limit of stationary Markov equilibria in \( \beta \) -discounted games (as \( \beta \rightarrow 1 \) ) is an equilibrium in the limit-average game. A related result was established by Borkar and Ghosh (199... | Yes |
Theorem 19. Every two-person stochastic game has a uniform equilibrium payoff. | The proof of Vrieze and Thuijsman (1989) is based on the \ | No |
Theorem 22. Every n-person stochastic game with finite state and action spaces has a uniform correlated equilibrium payoff using an autonomous correlation device. | The construction of an equilibrium profile is based on the method of Mertens and Neyman (1981) applied to zero-sum games. The equilibrium path is sustained by the use of threat strategies. However, punishment occurs only if a player disobeys the recommendation of the correlation device. | No |
Theorem 23. Every positive recursive stochastic game with finite sets of states and actions has a uniform correlated equilibrium payoff and the correlation device can be taken to be stationary. | The proof of the above result makes use of a variant of the method of Vieille (2000b). | No |
Theorem 25. Every intergenerational game (G1), (G2) and (G3) possesses a stationary Markov perfect equilibrium \( {c}^{ * } \in F \) . | The main idea of the proof is based upon the consideration of an operator \( L \) defined as follows: to each consumption strategy \( c \in F \) used by descendant (or descendants) the function \( L \) assigns the maximal element \( {c}_{0} \) from the set of best responses to \( c \) . It is shown that \( {c}_{0} \in ... | Yes |
Theorem 27. Assume (B3) and consider game (G3). Then, the set of invariant distributions \( \Delta \left( {q}^{ * }\right) \) is compact in the weak topology on \( \Pr \left( S\right) \) . | For each \( \sigma \in \Delta \left( {q}^{ * }\right), M\left( \sigma \right) \mathrel{\text{:=}} {\int }_{S}{s\sigma }\left( {ds}\right) \) is the mean of distribution \( \sigma \) . By Theorem 27, there exists \( {\sigma }^{* * } \) with the highest mean over the set \( \Delta \left( {q}^{ * }\right) \) . | No |
Theorem 28. In a non-paternalistic game as described above with nonatomic transitions, there exists a stationary equilibrium \( \left( {{c}^{ * },{v}^{ * }}\right) \in F \times V \) . | Theorem 28 was established as Theorem 1 in Balbus et al. (2016). Ray (1987) analysed games with non-paternalistic altruism and deterministic production functions. Unfortunately, his proof contains a mistake. The above result is strongly based on the assumption that the transitions are nonatomic and weakly continuous. T... | No |
Theorem 30. For each \( \epsilon > 0 \), the stopping game has an \( \epsilon \) -equilibrium \( \left( {{p}_{\epsilon }^{ * },{q}_{\epsilon }^{ * }}\right) \) . | Theorem 30 does not hold, if the payoffs are not uniformly bounded. Its proof is based upon a stochastic version of the Ramsey theorem that was also proved by Shmaya and Solan (2004). It states that for every colouring of a complete infinite graph by finitely many colours, there is a complete infinite monochromatic sub... | No |
Theorem 1 (Verification Theorem). Let us assume that the target is closed and that Isaacs’condition (8.2) holds. Suppose that there is a nonnegative map \( \mathbf{V} \) : \( {\mathbb{R}}^{d} \rightarrow \mathbb{R} \), continuous on \( {\mathbb{R}}^{d} \) and of class \( {\mathcal{C}}^{1} \) over \( {\mathbb{R}}^{d} \s... | Sketch of proof: Let us fix \( {x}_{0} \in {\mathbb{R}}^{d} \) . We first claim that\n\n\[ \mathop{\inf }\limits_{\bar{u}}{\theta }_{C}\left( {X}^{{x}_{0},\bar{u},{\bar{v}}^{ * }}\right) \geq V\left( {x}_{0}\right) \]\n\nFor this, let \( \bar{u} \in \bar{U} \) and let us set \( {X}_{t} = {X}^{{x}_{0},\bar{u},{\bar{v}}^... | Yes |
Theorem 2. Let \( {x}_{0} \in {\mathbb{R}}^{d} \smallsetminus C \) be an initial position and \( {\bar{u}}^{ * } \) and \( {\bar{v}}^{ * } \) be the optimal strategies given in Theorem (1) and let us set \( {X}_{t} = {X}_{t}^{{x}_{0},{\bar{u}}^{ * },{\bar{v}}^{ * }} \) . Then the pair \( \left( {X, P}\right) \) , where... | Sketch of proof: The starting point is the remark that one can express the derivative of \( H \) with respect to \( p \) in terms of \( \widetilde{u} \) and \( \widetilde{v} \) . Namely,\n\n\[ \n\frac{\partial H}{\partial p}\left( {x, p}\right) = f\left( {x,\widetilde{u}\left( {x, p}\right) ,\widetilde{v}\left( {x, p}\... | Yes |
Proposition 1. If \( \mathbf{V} \) is of class \( {\mathcal{C}}^{1} \) on \( \overline{{\mathbb{R}}^{d} \smallsetminus C} \), then\n\n\[ \forall x \in \partial C, H\left( {x,{v}_{x}}\right) < 0\text{ and }D\mathbf{V}\left( x\right) = - \frac{{v}_{x}}{H\left( {x,{v}_{x}}\right) }.\] | Indeed, if \( x \in \partial C \) and since \( \mathbf{V} = 0 \) on \( \partial C \) and is nonnegative on \( {\mathbb{R}}^{d} \smallsetminus C \), it has a minimum on \( \overline{{\mathbb{R}}^{d} \smallsetminus C} \) at \( x \) . The Karush-Kuhn-Tucker condition then implies the existence of \( \lambda \geq 0 \) with... | Yes |
Proposition 2. If \( x \in {UP} \), then \( \mathbf{V} \) is continuous and vanishes at \( x \) . On the contrary, if \( H\left( {x,{v}_{x}}\right) > 0 \), then \( \mathbf{V} \) is bounded below by a positive constant in a neighborhood of \( x \) in \( {\mathbb{R}}^{d} \smallsetminus C \) . | Sketch of proof: If \( x \in {UP} \), then, from the definition of \( H \) , \[ H\left( {x,{v}_{x}}\right) = \mathop{\inf }\limits_{{u \in U}}\mathop{\sup }\limits_{{v \in V}}\left\langle {f\left( {x, u, v}\right) ,{v}_{x}}\right\rangle < 0. \] Thus, there exists \( {u}_{0} \in U \) such that, for any \( v \in V,\left\... | No |
Proposition 3. In the above configuration, one has \( H\left( {x,{D\psi }\left( x\right) }\right) = 0 \) . | Sketch of proof: If on the contrary one had \( \;H\left( {x,{D\psi }\left( x\right) }\right) > 0 \), then, from the very definition of \( H \), there would exist \( {v}_{0} \in V \) for the second player such that\n\n\[ \mathop{\inf }\limits_{{u \in U}}\left\langle {f\left( {x, u,{v}_{0}}\right) ,{D\psi }\left( x\right... | No |
Lemma 1. Let \( \alpha \in \mathcal{A}\left( {t}_{0}\right) \) and \( \beta \in \mathcal{B}\left( {t}_{0}\right) \) . Assume that either \( \alpha \) or \( \beta \) is a delay strategy. Then there is a unique pair of controls \( \left( {u, v}\right) \in {U}_{{t}_{0}} \times {V}_{{t}_{0}} \) such that\n\n\[ \alpha \left... | Sketch of proof: Let us assume, to fix the ideas, that \( \alpha \) is a delay strategy with delay \( \tau \) . We first note that the restriction of \( u \mathrel{\text{:=}} \alpha \left( v\right) \) to the interval \( \left\lbrack {{t}_{0},{t}_{0} + \tau }\right\rbrack \) is independent of \( v \) because any two con... | No |
Theorem 3. Assume that (8.9) and (8.10) hold, as well as Isaacs' condition:\n\n\[ \mathop{\inf }\limits_{{u \in U}}\mathop{\sup }\limits_{{v \in V}}\{ \langle p, f\left( {t, x, u, v}\right) \rangle + \ell \left( {t, x, u, v}\right) \} = \mathop{\sup }\limits_{{v \in V}}\mathop{\inf }\limits_{{u \in U}}\{ \langle p, f\l... | Remark 2. From the very definition of the value functions, one easily checks that \( {\mathbf{V}}^{ - } \leq {\mathbf{V}}^{ + } \) . So the key point is to prove the reverse inequality. The proof is not easy. It consists in showing that:\n\n1. both value functions satisfy a dynamic programming property and enjoy some w... | No |
Proposition 4 (see Bardi and Capuzzo Dolcetta (1996)). Let \( {\mathbf{V}}^{ - } \) and \( {\mathbf{V}}^{ + } \) be the lower and upper value functions as in Definition 5. Then\n\n\[ \n{\mathbf{V}}^{ - }\left( {{t}_{0},{x}_{0}}\right) = \mathop{\inf }\limits_{{\alpha \in \mathcal{A}\left( {t}_{0}\right) }}\mathop{\sup ... | The proof of the proposition involves a dynamic programming for the value functions defined in the proposition in terms of nonanticipative strategies and derive from this that they satisfy the same HJI equation than \( {\mathbf{V}}^{ + } \) and \( {\mathbf{V}}^{ - } \) . Whence the equality. | No |
Theorem 4 (Dynamic programming). Let \( \left( {{t}_{0},{x}_{0}}\right) \in \lbrack 0, T) \times {\mathbb{R}}^{d} \) and \( h \in (0 \) , \( \left. {T - {t}_{0}}\right) \) . Then\n\n\[ \n{\mathbf{V}}^{ + }\left( {{t}_{0},{x}_{0}}\right) = \mathop{\inf }\limits_{{\alpha \in {\mathcal{A}}_{d}\left( {t}_{0}\right) }}\math... | Sketch of proof: Following Remark 1, we have\n\n\[ \n{\mathbf{V}}^{ + }\left( {{t}_{0},{x}_{0}}\right) = \mathop{\inf }\limits_{{\alpha \in {\mathcal{A}}_{d}\left( {t}_{0}\right) }}\mathop{\sup }\limits_{{v \in {V}_{{t}_{0}}}}\left\{ {{\int }_{{t}_{0}}^{T}\ell \left( {{X}_{s}^{{t}_{0},{x}_{0},\alpha \left( v\right), v}... | Yes |
Theorem 5 (Comparison principle, Crandall et al. (1992)). Under assumptions (8.22) and (8.23), let \( {\mathbf{V}}_{1} \) be a subsolution of (8.21) which is u.s.c. on \( \left\lbrack {0, T}\right\rbrack \times {\mathbb{R}}^{d} \) and \( {\mathbf{V}}_{2} \) be a supersolution of (8.21) which is l.s.c. on \( \left\lbrac... | \[ {\mathbf{V}}_{1}\left( {t, x}\right) \leq {\mathbf{V}}_{2}\left( {t, x}\right) \;\forall \left( {t, x}\right) \in \left\lbrack {0, T}\right\rbrack \times {\mathbb{R}}^{d}. \] | Yes |
Corollary 1. Let \( g : {\mathbb{R}}^{d} \rightarrow \mathbb{R} \) be continuous. Then Eq. (8.21) has at most one continuous viscosity solution which satisfies the terminal condition \( \mathbf{V}\left( {T, x}\right) = g\left( x\right) \) for any \( x \in {\mathbb{R}}^{d} \) . | Proof. Let \( {\mathbf{V}}_{1} \) and \( {\mathbf{V}}_{2} \) be two bounded and Lipschitz continuous viscosity solutions of (8.21) such that \( {\mathbf{V}}_{1}\left( {T, x}\right) = {\mathbf{V}}_{2}\left( {T, x}\right) = g\left( x\right) \) for any \( x \in {\mathbb{R}}^{d} \) . Since, in particular, \( {\mathbf{V}}_{... | Yes |
Theorem 6. Under conditions (8.9) and (8.10) on \( f,\ell \), and \( g \) and if Isaacs’ assumption holds,\n\n\[ \n{H}^{ + }\left( {t, x, p}\right) = {H}^{ - }\left( {t, x, p}\right) \;\forall \left( {t, x, p}\right) \in \left\lbrack {0, T}\right\rbrack \times {\mathbb{R}}^{d} \times {\mathbb{R}}^{d}, \n\]\n\n(8.24)\n\... | The key point of the proof of Theorem 6 is the following (half-)characterization of the value functions:\n\nLemma 2. The upper and | No |
Lemma 2. The upper and lower value functions \( {\mathbf{V}}^{ + } \) and \( {\mathbf{V}}^{ - } \) are respectively viscosity solutions of equation (8.18) and of (8.19). | The proof, which is a little intricate, follows more or less the formal argument described in Sect. 3.3. Let us underline at this point that if Isaacs' condition does not hold, there is no reason in general that the upper value function and the lower one coincide. Actually they do not in general, in the sense that ther... | No |
Proposition 6. The lower value function \( {\mathbf{V}}^{ - } \) is a viscosity subsolution of the HJI equation\n\n\[ \left\{ \begin{matrix} - {\partial }_{t}\mathbf{V} - {H}^{ - }\left( {t, x, D\mathbf{V},{D}^{2}\mathbf{V}}\right) = 0,\left( {t, x}\right) \in \left( {0, T}\right) \times {\mathbb{R}}^{d}, \\ \mathbf{V}... | The proof of the proposition is extremely tricky and technical: a sub-dynamic programming principle for the lower value function can be obtained, provided that the set of strategies for player 1 is replaced by a smaller one, where some additional measurability condition is required. From this sub-dynamic programming pr... | No |
Proposition 8 (Geometric formulation). Let \( {\operatorname{Ker}}_{f}\left( K\right) \) be the discriminating kernel of \( K \) . For any \( x \in \overline{K \smallsetminus {Ke}{r}_{f}\left( K\right) } \) which lies in the interior of \( K \) and for any \( v \) proximal normal to \( \overline{K \smallsetminus {\oper... | As explained below, this inequality formally means that player 2 can prevent the state of the system to enter the interior of the discriminating kernel. Recalling that the discriminating kernel is a discriminating domain, thus satisfying the geometric condition (8.48), one concludes that the boundary of the victory dom... | No |
Theorem 12. Under the above assumptions, the game has a value: \( \mathbf{V} \mathrel{\text{:=}} {\mathbf{V}}^{ + } = {\mathbf{V}}^{ - } \) , which is continuous on its domain \( \operatorname{dom}\left( \mathbf{V}\right) \) and is the unique viscosity of the following HJI equation:\n\n\[ \left\{ \begin{array}{l} - H\l... | Sketch of proof: Thanks to the controllability condition, one can check that the \( {\mathbf{V}}^{ \pm } \) are continuous on \( \partial C \) and vanish on this set. The next step involves using the regularity of the dynamics and the cost to show that the domains of the \( {\mathbf{V}}^{ \pm } \) are open and that \( ... | No |
Theorem 14 (Alpern-Gal ('88)). The game has a value:\n\n\[ \mathop{\inf }\limits_{{\mu \in \Delta \left( \mathcal{X}\right) }}\mathop{\sup }\limits_{{v \in \Delta \left( \mathcal{Y}\right) }}{\int }_{\mathcal{X} \times \mathcal{Y}}\tau \left( {X, Y}\right) {d\mu }\left( X\right) {dv}\left( Y\right) \]\n\n\[ = \mathop{\... | Moreover, there exists an optimal strategy \( \bar{\mu } \) for the pursuer. | Yes |
Theorem 15 (Gal (’79,’80)). Assume that \( Q \) is a compact convex subset of \( {\mathbb{R}}^{2} \) , and let \( v\left( r\right) \) be the value of the game for the radius \( r \) . Then, as \( r \rightarrow {0}^{ + } \) , | \[ v\left( r\right) \sim \frac{\left| Q\right| }{2r} \] where \( \left| Q\right| \) is the Lebesgue measure of \( Q \) . | Yes |
Theorem 16. Under the above assumption, the game has a value: \( {\mathbf{V}}^{ + } = {\mathbf{V}}^{ - } \) . This value \( \mathbf{V} \mathrel{\text{:=}} {\mathbf{V}}^{ + } = {\mathbf{V}}^{ - } \) is convex with respect to the \( p \) variable and solves in the viscosity sense the Hamilton-Jacobi equation\n\n\[ \left\... | Sketch of proof. The main issue is that the dynamic principle does not apply in a standard way: indeed, the non-informed player (i.e., player 2) might learn at a part of his missing information by observing his opponent's control. So one cannot start afresh the game without taking into account the fact that the informa... | Yes |
Theorem 17. The following equality holds:\n\n\[ \mathbf{V}\left( {{t}_{0},{p}_{0}}\right) = \min \mathbb{E}\left\lbrack {{\int }_{{t}_{0}}^{T}H\left( {s,\mathbf{p}\left( s\right) }\right) {ds}}\right\rbrack \;\forall \left( {{t}_{0},{p}_{0}}\right) \in \left\lbrack {0, T}\right\rbrack \times \Delta \left( I\right) ,\] | where the minimum is taken over all the (càdlàg) martingales \( \mathbf{p} \) on \( \Delta \left( I\right) \) starting from \( {p}_{0} \) at time \( {t}_{0} \) .\n\nThe martingale process can be interpreted as the information on the index \( i \) the informed player discloses along the time. One can show that the infor... | Yes |
Assume that \( H = H\left( p\right) \) does not depend on \( t \) . Then we claim that\n\n\[ \mathbf{V}\left( {t, p}\right) = \left( {T - t}\right) \operatorname{Vex}H\left( p\right) \;\forall \left( {t, p}\right) \in \left\lbrack {0, T}\right\rbrack \times \Delta \left( I\right) ,\] \n\nwhere \( \operatorname{Vex}H\le... | Indeed let us set \( W\left( {t, p}\right) = \) \( \left( {T - t}\right) \operatorname{Vex}H\left( p\right) \) . Then (at least in a formal way), \( W \) is convex in \( p \) and satisfies\n\n\[ - {\partial }_{t}W\left( {t, p}\right) - H\left( p\right) \leq - {\partial }_{t}W\left( {t, p}\right) - \operatorname{Vex}H\l... | No |
Assume that \( I = 2 \), and let us identify \( p \in \left\lbrack {0,1}\right\rbrack \) with the pair \( \left( {p,1 - p}\right) \in \Delta \left( 2\right) \) . We suppose that there exists \( {h}_{1},{h}_{2} : \left\lbrack {0, T}\right\rbrack \rightarrow \left\lbrack {0,1}\right\rbrack \) continuous, \( {h}_{1} \leq ... | Then one can show that \( \mathbf{V} \) can be explicitly computed:\n\n\[ \mathbf{V}\left( {t, p}\right) = {\int }_{t}^{T}\operatorname{Vex}H\left( {s, p}\right) {ds}\;\forall \left( {t, p}\right) \in \left\lbrack {0, T}\right\rbrack \times \Delta \left( I\right) .\n\nMoreover, the optimal martingale is unique and can ... | Yes |
Theorem 46 (Lions-Papanicolaou-Varadhan ('86)). Under the above assumptions, there exists a constant \( \bar{c} \in \mathbb{R} \) such that\n\n\[ \mathop{\lim }\limits_{{T \rightarrow + \infty }}\frac{{\mathbf{V}}_{T}\left( {0, x}\right) }{T} = \mathop{\lim }\limits_{{lambda \rightarrow {0}^{ + }}}\lambda {\mathbf{V}}_... | Sketch of proof. We start with \( {\mathbf{V}}_{\lambda } \) and first note that \( {\mathbf{V}}_{\lambda } \) is \( {\mathbb{Z}}^{d} \) -periodic. We also note that, thanks to the maximum principle and Eq. (8.56),\n\n\[ \mathop{\inf }\limits_{{x \in {\mathbb{R}}^{d}}}H\left( {x,0}\right) \leq \lambda {\mathbf{V}}_{\la... | Yes |
Theorem 47 (Lions-Papanicolaou-Varadhan (’86)). As \( \epsilon \rightarrow 0,{\mathbf{V}}^{\epsilon } \) converges locally uniformly to \( \overline{\mathbf{V}} \) solution to\n\n\[ \left\{ \begin{array}{l} - {\partial }_{t}\overline{\mathbf{V}}\left( {t, x, y}\right) - \bar{H}\left( {x,{D}_{x}\overline{\mathbf{V}},{D}... | The heuristics behind the result is the following. Let us assume that \( {\mathbf{V}}^{\epsilon } \) uniformly converges to a map \( \overline{\mathbf{V}} \) . In order to guess the equation satisfied by \( \overline{\mathbf{V}} \), let us try to write a first-order expansion in \( \epsilon \) of \( {\mathbf{V}}^{\epsi... | Yes |
Theorem 1. If the auxiliary problem admits one or several solutions leading to a unique state \( \widehat{x}\left( t\right) \) at time \( t \), then, if the controller\n\n\[ u\left( t\right) = {\varphi }^{ \star }\left( {t,\widehat{x}\left( t\right) }\right) \]\n\nis admissible, it is a min-sup controller in partial in... | One way to proceed in the case of an information such as (9.7) is to solve by forward dynamic programming (forward Hamilton-Jacobi-Caratheodory-Bellman equation) the constrained problem:\n\n\[ \mathop{\max }\limits_{\omega }\left\lbrack {{\int }_{{t}_{0}}^{t}L\left( {s, x\left( s\right), u\left( s\right), w\left( s\rig... | Yes |
Theorem 3. For a given \( \gamma \), if both equations (9.23) and (9.24) have solutions over \( \left\lbrack {{t}_{0},{t}_{1}}\right\rbrack \) satisfying\n\n\[ \forall t \in \left\lbrack {{t}_{0},{t}_{1}}\right\rbrack ,\;\rho \left( {\sum \left( t\right) S\left( t\right) }\right) \leq {\gamma }^{2}, \] \n\n(9.25)\n\nth... | \[ \dot{\widehat{x}} = \left( {A + {\gamma }^{-2}{MS}}\right) \widehat{x} + B{u}^{ \star } + {\left( I - {\gamma }^{-2}\sum S\right) }^{-1}K\left\lbrack {y - \left( {C + {\gamma }^{-2}{LS}}\right) \widehat{x}}\right\rbrack ,\widehat{x}\left( {t}_{0}\right) = 0, \] \n\n\[ {u}^{ \star } = F\widehat{x} \] \n\nor \n\n\[ \d... | Yes |
Theorem 5. Under the hypothesis 3, if both discrete Riccati equations (9.34) and (9.35) have solutions satisfying either \( \rho \left( {{M}_{t}{S}_{t + 1}}\right) < {\gamma }^{2} \) and \( \rho \left( {{\widetilde{\sum }}_{t + 1}{S}_{t + 1}}\right) < {\gamma }^{2} \) or \( \rho \left( {{\sum }_{t}{Q}_{t}}\right) < {\g... | \[ {u}_{t}^{ \star } = - {R}_{t}^{-1}\left( {{B}_{t}^{\prime }{\Gamma }_{t}{\bar{A}}_{t} + {P}_{t}^{\prime }}\right) {\left( I - {\gamma }^{-2}{\sum }_{t}{S}_{t}\right) }^{-1}{\check{x}}_{t}, \] (9.36) \[ {\check{x}}_{t + 1} = {A}_{t}{\check{x}}_{t} + {B}_{t}{u}_{t}^{ \star } + {\gamma }^{-2}{\widetilde{A}}_{t}{\Delta ... | Yes |
Theorem 10. The Value function associated with the differential game defined by equations (9.48) (9.49), and (9.54) is the unique Lipshitz continuous viscosity solution of the differential variational inequality (9.55). | Solving the DQVI (9.55) may be done with the help of the following auxiliary functions. We define\n\n\[ \n{q}^{ - }\left( t\right) = \max \left\{ {\left( {1 + {c}^{ - }}\right) \exp \left\lbrack {{\tau }^{ - }\left( {T - t}\right) }\right\rbrack - 1,{C}^{ - }}\right\} , \n\]\n\n\[ \n{q}^{ + }\left( t\right) = \min \lef... | Yes |
Theorem 12. The Value function \( \left\{ {V}_{k}^{h}\right\} \) satisfies the Isaacs recurrence equation: | \[ {V}_{k}^{h}\left( {u, v}\right) = \mathop{\min }\limits_{\xi }\mathop{\max }\limits_{{\tau \in \left\lbrack {{\tau }_{h}^{ + },{\tau }_{h}^{ + }}\right\rbrack }}\left\lbrack {{V}_{k + 1}^{h}\left( {\left( {1 + \tau }\right) u,\left( {1 + \tau }\right) \left( {v + \xi }\right) }\right) - \tau \left( {v + \xi }\right)... | Yes |
Theorem 3. Suppose patch fitness is a decreasing function of patch density in a single-species habitat selection game. Then any migration dynamics (10.9) that satisfies the following two conditions evolves to the unique IFD.\n\n(a) Individuals never move to a patch with lower fitness.\n\n(b) If there is a patch with hi... | \( {}^{12} \) To see that the habitat selection game is a potential game, take \( F\left( p\right) \equiv \mathop{\sum }\limits_{{i = 1}}^{H}{\int }_{0}^{{p}_{i}}\pi \left( {{e}_{i},{u}_{i}}\right) d{u}_{i} \) . Then \( \frac{\partial F\left( p\right) }{\partial {p}_{i}} = \pi \left( {{e}_{i},{p}_{i}}\right) \) . If pa... | Yes |
Theorem 2. Suppose that \( S \) is one dimensional and \( {u}^{ * } \in \operatorname{int}\left( S\right) \) is a rest point of adaptive dynamics (10.13) (i.e., \( {\pi }_{1}\left( {{u}^{ * },{u}^{ * }}\right) = 0 \) ).\n\n(a) \( {u}^{ * } \) is an NIS and a neighborhood strict \( {NE} \) if and only if it is neighborh... | The proof of Theorem 2 relies on results based on the Taylor expansion (10.14). For instance, along with the characterizations of a strict NE as \( {\pi }_{11} < 0 \) and convergence stability as \( {\pi }_{11} + {\pi }_{12} < 0 \) from Sect. 3.1, \( {u}^{ * } \) is an NIS if and only if \( \frac{1}{2}{\pi }_{11} + {\p... | Yes |
Theorem 5 (Maximum Principle). Suppose that \( \left( {{\mathbf{u}}^{ * },{\mathbf{n}}^{ * }}\right) \) is an asymptotically stable rest point for Darwinian dynamics (10.16) and (10.17) applied to a resident system. If \( \left( {{\mathbf{u}}^{ * },{\mathbf{n}}^{ * }}\right) \) is a stable evolutionary outcome, then\n\... | This fundamental result promoted by Vincent and Brown (see, for instance, their 2005 book) gives biologists the candidate solutions they should consider when looking for stable evolutionary outcomes to their biological systems. That is, by plotting the G-function as a function of \( v \) for a fixed candidate \( \left(... | Yes |
Theorem 6. Suppose \( {u}^{ * } \in \operatorname{int}\left( S\right) \) is a rest point of (10.23) (i.e., \( {\nabla }_{1}\pi \left( {{u}^{ * },{u}^{ * }}\right) = 0 \) ). | (a) \( {u}^{ * } \) is a neighborhood strict \( {NE} \) if and only if \( A \) is negative definite. It is convergence stable for all choices of \( {C}_{1}\left( u\right) \) if and only if \( A + B \) is negative definite. It is a CSS if and only if it is neighborhood half-superior if and only if it is a neighborhood s... | Yes |
Example 2 (Two-species habitat selection game). Suppose that there are two species competing in two different habitats (or patches) and that the overall population size (i.e., density) of each species is fixed. Also assume that the fitness of an individual depends only on its species, the patch it is in, and the densit... | To see this result first proven by Selten (1980), take \( \left( {p, q}\right) = \left( {p,{q}^{ * }}\right) \) . Then (10.25) implies \( p \cdot A{q}^{ * } < {p}^{ * } \cdot A{q}^{ * } \) or \( {q}^{ * } \cdot B{p}^{ * } < {q}^{ * } \cdot B{p}^{ * } \) for all \( p \neq {p}^{ * } \) . Thus, \( p \cdot A{q}^{ * } < {p}... | No |
What are the Nash equilibria (NEs) for this example? | If players 1 and 2 choose \( R \) and \( r \), respectively, with payoff 2 for both, then\n\n1. player 2 does worse through unilaterally changing his strategy by playing \( r \) with probability \( 1 - q \) less than 1 (since \( {0q} + 2\left( {1 - q}\right) < 2 \) ) and\n\n2. player 1 does worse through unilaterally c... | Yes |
Can evolutionary dynamics be used to select one of the two NE outcomes of the Chain Store game? Suppose players 1 and 2 use mixed strategies \( p \) and \( q \), respectively. The payoffs of pure strategies \( L \) and \( R \) for player 1 (denoted \( {\pi }_{1}\left( {L, q}\right) \) and \( {\pi }_{1}\left( {R, q}\rig... | Under the replicator equation, the probability of using a pure strategy increases if its payoff is higher than these expected payoffs. For this example, the replicator equation is (Weibull 1995, see also Remark 5 below)\n\n\[ \dot{p} = p\left( {1 - \left( {p + \left( {1 - p}\right) 2\left( {1 - q}\right) }\right) }\rig... | Yes |
Consider the well-studied consensus/rendezvous problem where the goal is to drive the agents to agreement on a state \( {x}^{ * } \in \mathbb{R} \) when each agent has limited information regarding the state of other agents in the systems. Specifically, we will say that the set of admissible states (or actions) of each... | Now we will present a game-theoretic design that leads to the same collective behavior. More formally, consider a game-theoretic model where each agent \( i \in N \) is assigned an action set \( {\mathcal{A}}_{i} = \mathbb{R} \) and a utility function of the form\n\n\[ \n{U}_{i}\left( {{a}_{i},{a}_{-i}}\right) = - \fra... | Yes |
Consider any game \( G \) where the agents’ utility functions satisfy \( \mathop{\sum }\limits_{{i \in N}}{U}_{i}\left( a\right) \leq W\left( a\right) \) for any \( a \in \mathcal{A} \) . If there exists parameters \( \lambda > 0 \) and \( \mu > - 1 \) such that for any two action profiles \( a,{a}^{ * } \in \mathcal{A... | \[ \mathop{\sum }\limits_{i}{U}_{i}\left( {{a}_{i}^{ * },{a}_{-i}}\right) \geq \lambda \cdot W\left( {a}^{ * }\right) - \mu \cdot W\left( a\right) ,\] then the efficiency associated with any coarse correlated equilibrium \( {z}^{\text{cce }} \in \Delta \left( \mathcal{A}\right) \) of \( G \) must satisfy \[ \frac{W\lef... | Yes |
Theorem 2 (Roughgarden 2015; Vetta 2002). Consider any game \( G = \) \( \left( {N,\left\{ {\mathcal{A}}_{i}\right\} ,\left\{ {U}_{i}\right\}, W}\right) \) that satisfies the following three properties:\n\n(i) The objective function \( W \) is submodular;\n\n(ii) For any agent \( i \in N \) and any action profile \( a ... | One example of a valid utility game is the vehicle-target assignment problem which will be presented in Example 4. Here, the system-level objective function is submodular and Condition (ii) in Theorem 2 is satisfied by the given design. Further, it is straightforward to verify that Condition (iii) is also satisfied. Ac... | Yes |
A routing problem consists of a collection of self-interested agents that need to utilize a common network to satisfy their individual demands. The network is characterized by a collection of edges \( E = \) \( \left\{ {{e}_{1},\ldots ,{e}_{m}}\right\} \) where each edge \( e \in E \) is associated with an anonymous co... | It is well known that any routing game of the above form, which is commonly referred to as an anonymous congestion game, is a potential game with a potential function \( \phi : \mathcal{A} \rightarrow \mathbb{R} \) of the form \[ \phi \left( a\right) = \mathop{\sum }\limits_{{e \in E}}\mathop{\sum }\limits_{{k = 1}}^{{... | Yes |
In the case where the common action set has two actions, i.e., \( \overline{\mathcal{A}} = \{ x, y\} \), and the interaction graph is undirected, i.e., \( j \in {\mathcal{N}}_{i} \Leftrightarrow i \in {\mathcal{N}}_{j} \), it is straightforward to show that this utility structure gives rise to a potential game with a p... | \[ \phi \left( a\right) = \frac{1}{2}\mathop{\sum }\limits_{{\left( {i, j}\right) \in E}}{\phi }_{\mathrm{{pw}}}\left( {{a}_{i},{a}_{j}}\right) \] where \( {\phi }_{\mathrm{{pw}}} : \overline{\mathcal{A}} \times \overline{\mathcal{A}} \rightarrow \mathbb{R} \) is a local potential function. One choice for this local po... | Yes |
In the well-studied vehicle-target assignment problem, there is a finite set of targets \( \mathcal{T} \), and each target \( t \in \mathcal{T} \) has a relative value of importance \( {v}_{t} \geq 0 \) . Further, there are a set of vehicles \( N = \) \( \{ 1,2,\ldots, n\} \) where each vehicle \( i \in N \) has an inv... | Consider one such design where the utility functions of the vehicles are set as the marginal contribution of the vehicles to the system-level objective, i.e., for each vehicle \( i \in N \) and allocation \( a \in \mathcal{A} \) we have\n\n\[ \n{U}_{i}\left( a\right) = \mathop{\sum }\limits_{{t \in {a}_{i}}}{v}_{t} \cd... | Yes |
Theorem 5. Consider any exact potential game \( G \) . If all players following the learning algorithm joint strategy fictitious play defined above, then the joint action profile will converge almost surely to a pure Nash equilibrium of the game \( G \) . | Hence, JFSP with inertia provides similar asymptotic guarantees to fictitious play while minimizing the computational and observational burden on the agents. The name \ | No |
Theorem 7. Consider any potential game \( G \) with potential function \( \phi \) . If all players follow the learning algorithm log-linear learning with temperature \( T > 0 \), then the resulting process has a unique stationary distribution \( \pi = {\left\{ {\pi }^{a}\right\} }_{a \in \mathcal{A}} \in \Delta \left( ... | The stationary distribution of the process given in (11.29) follows the same intuition as presented for the update protocol in (11.28). That is, when \( T \rightarrow \infty \) the stationary distribution is effectively a uniform distribution over the joint action set \( \mathcal{A} \) . However, when \( T \rightarrow ... | Yes |
Consider the class of resource allocation problems defined in Sect. 2.1 with agent set \( N \), action sets \( \left\{ {\mathcal{A}}_{i}\right\} \), and a global objective \( W : \mathcal{A} \rightarrow \mathbb{R} \) . Consider the following game-theoretic control design:\n\n(i) Assign each agent a utility function tha... | Observe that this design rule ensures that the resulting asymptotic behavior will be concentrated around the allocations that maximize the global objective \( W \) . This fact has made this design methodology an attractive option for several domains including wind farms, sensor networks, and coordination of unmanned ve... | Yes |
Theorem 8. Consider any potential game \( G \) with potential function \( \phi \) . If all players follow the learning algorithm binary log-linear learning with restricted action set and temperature \( T > 0 \), then an action profile is stochastically stable if and only if it is a potential function maximizer. | This theorem demonstrates that a system designer can effectively deal with restrictions in action sets by appropriately modifying the learning rule. However, a consequence of this is that we are no longer able to provide a precise characterization of the stationary distribution as a function of the temperature paramete... | No |
Theorem 9. Consider any finite game \( G \) . If all players follow the learning algorithm regret matching defined above, then the positive regret for any agent \( i \in N \) and action \( {a}_{i} \in {\mathcal{A}}_{i} \) asymptotically vanishes, i.e., | \[ \mathop{\lim }\limits_{{t \rightarrow \infty }}{\left\lbrack {R}_{i}^{{a}_{i}}\left( t\right) \right\rbrack }_{ + } = 0 \] Alternatively, the empirical frequency of play converges to the set of coarse correlated equilibria. The connection between the condition (11.36) and the definition of coarse correlated equilibr... | Yes |
Example 1 (Network congestion games). Let \( \mathcal{G} = \left( {\mathcal{V},\mathcal{E}}\right) \) be a directed network with vertex set \( \mathcal{V} \) and edge set \( \mathcal{E} \), such that each player \( i \in \left\lbrack n\right\rbrack \) is associated with a pair of nodes \( \left( {{s}_{i},{t}_{i}}\right... | In fact, congestion games (and in particular network congestion games) feature many nice properties, perhaps the most remarkable one being that they admit at least one pure-strategy Nash equilibrium (NE), \( {}^{3} \) which is mainly due to their structure. More generally, congestion games are known to belong to the cl... | Yes |
Theorem 1. Any exact potential game is isomorphic to a congestion game. | The main idea behind establishing the result of Theorem 1 is to show that for a game with an exact potential function \( \Phi \left( \cdot \right) \), one can construct an equivalent congestion game in which the players are the same, and the resources are interpreted as all possible subsets of actions which can be take... | No |
Theorem 2. There is a polynomial algorithm for finding a pure-strategy \( {NE} \) in symmetric network congestion games. | This result is established by showing that for the symmetric network congestion games, finding an action profile which minimizes the potential function \( \Phi \left( \mathbf{a}\right) = \) \( \mathop{\sum }\limits_{{e \in \mathcal{E}}}\mathop{\sum }\limits_{{j = 1}}^{{{n}_{e}\left( a\right) }}{d}_{e}\left( j\right) \)... | Yes |
Theorem 3. Market sharing game is a special case of congestion games and hence admits a pure-strategy NE. | The above theorem simply holds by noting that the Rosenthal potential function \( \Phi \left( \mathbf{a}\right) = \mathop{\sum }\limits_{{j = 1}}^{m}\mathop{\sum }\limits_{{k = 1}}^{{n}_{j}}\frac{{q}_{j}}{k} \) adapted for the market sharing game serves as an exact potential function for this game. In fact, in the mark... | Yes |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.