SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Ioannis Panageas*
University of California, Irvine ipanagea@ics.uci.edu
Nikolas Patris*
University of California, Irvine npatris@uci.edu
Fictitious Play (FP) is a simple and natural dynamic for repeated play with many applications in game theory and multi-agent reinforcement learning. It was introduced by Brown [4, 5] and its convergence properties for two-player zero-sum games was established later by Robinson [22]. Potential games [19] is another class of games which exhibit the FP property [18], i.e., FP dynamics converges to a Nash equilibrium if all agents follows it. Nevertheless, except for two-player zero-sum games and for specific instances of payoff matrices [1] or for adversarial tie-breaking rules [9], the convergence rate of FP is unknown. In this work, we focus on the rate of convergence of FP when applied to potential games and more specifically identical payoff games. We prove that FP can take exponential time (in the number of strategies) to reach a Nash equilibrium, even if the game is restricted to two agents and for arbitrary tie-breaking rules. To prove this, we recursively construct a two-player coordination game with a unique Nash equilibrium. Moreover, every approximate Nash equilibrium in the constructed game must be close to the pure Nash equilibrium in  $\ell_1$ -distance.
In 1949 Brown [4, 5] proposed an uncoupled dynamics called fictitious play so as to capture the behavior of selfish agents engaged in a repeatedly-played game. Fictitious play assumes at round  $t = 1$ , each agent selects an arbitrary action. At each round  $t \geq 2$ , each player plays a best response pure action to the opponents' empirical strategy; empirical strategy is defined to be the average of the past chosen strategies.
Due to its simplicity and natural behavioral assumptions, fictitious play is one of the most seminal and well-studied game dynamics [7]. Despite the fact that fictitious play does not converge to a Nash equilibrium (NE) in general normal-form games, there are several important classes of games at which the empirical strategies always converge to a NE. In her seminal work, Robinson [22] showed that in the case of two-player zero-sum games, the empirical strategy profiles converge to a min-max equilibrium of the game; Robinson's proof use a smart inductive argument on the number of strategies of the game. Later, Monderer and Shapley [18] established that in the case of  $N$ -player potential games, the empirical strategies also converge to a NE. Summing up, the following theorem is true:
Informal Theorem [22],[18] In two-player zero-sum and  $N$ -player potential games, the empirical strategy profiles of fictitious play converge to a NE for any initialization and tie-breaking rule*.
The above convergence results are asymptotic in the sense that they do not provide guarantees on the number of rounds needed by fictitious play to reach an approximate NE. Karlin [13] conjectured that in the case of two-player zero-sum games, fictitious play requires  $O(1 / \epsilon^2)$  rounds to reach an  $\epsilon$ -approximate NE. Daskalakis et. al. [9] disproved the strong version of Karlin's conjecture by providing an adversarial tie-breaking rule for which fictitious play requires exponential number of rounds (with respect to the number of strategies) in order to converge to an  $\epsilon$ -approximate NE. On the other hand, for the case of potential games and despite the foundational work of Monderer et al. [18], no lower bound results have been established. Therefore, the central focus of our work is to address the following question."
Q: Does fictitious play in potential games admit convergence to an approximate NE with rates that depend polynomially on the number of actions and the desired accuracy?
Our contributions Our work provides a negative answer on the above question. Specifically, we present a two-player potential game for which fictitious play requires super-exponential time with respect to the number of actions to reach an approximate NE.
Theorem 1.1 (Main result, formally stated Theorem 3.1). There exists a two-player potential game (more specifically both agents have identical payoffs) in which both agents admit  $n$  actions and for which fictitious play requires  $\Omega \left(4^n\left(\left(\frac{n}{2} - 2\right)! \right)^4 + \frac{1}{n\sqrt{\epsilon}} \right)$  rounds in order to reach an  $\epsilon$ -approximate NE. Moreover, the result holds for any tie-breaking rule and uniformly random initialization.
Remark 1.2. Daskalakis et al. [9] provide an exponential lower bound on the convergence of fictitious play assuming an adversarial tie-breaking rule, meaning that ties are broken in favor of slowing down the convergence rate. To this point, it is not known whether fictitious play with a consistent tie-breaking rule (e.g. lexicographic) converges in polynomial time or not, except for special cases, e.g., diagonal payoff matrices [1]. We would like to note that our lower bound construction holds for any tie-breaking rule. The latter indicates an interesting discrepancy between two-player zero-sum and two-player potential games.
Related Work The work of Daskalakis et al. [9] provides a lower bound on the convergence rate of fictitious play for the case of two-player zero-sum games using an adversarial tie-breaking rule and is the most closely related to our. Another important work is [1], in which they demonstrate that when the tie-breaking rule is predetermined (e.g., using lexicographical tie-breaking), the convergence rate of fictitious play becomes polynomial in the number of strategies, specifically with a rate of  $O\left(\frac{1}{\epsilon^2}\right)$ , as Karlin's conjecture states. It is important to note that this result holds true only for diagonal matrices.
Other works include the examination of fictitious play convergence for near-potential games [6], and the investigation of the necessary conditions that games must meet for fictitious play to converge to a NE [8]. We note that neither of these works provide convergence rates for fictitious play. On the other hand, fast convergence rates of continuous time fictitious play for regular, as introduced by Harsanyi [10], potential games are established [25]. Additional works on continuous-time fictitious play can be found in [21], and references there in.
Fictitious play has found various applications in Multi-agent Reinforcement Learning as well (see [20, 23, 2, 3] and [26] for a survey and references therein), extensive form games [11], and control theory [15, 14]. It possesses not only an intuitive game theoretic interpretation but also other attractive properties, such as its simplicity with no need for a step-size parameter. These traits have contributed to its status as a pivotal topic of study, leading to numerous works aiming to establish convergence in broader settings [16, 24, 17, 12].*
Technical overview The technical overview of this paper provides a high-level roadmap of the key contributions. In Section 3, we outline the steps towards proving Theorem 3.1 by recursively constructing a payoff matrix of carefully crafted structural properties. In that matrix, starting from the
* Tie-breaking rule when selecting between two or more different best-response actions.
*Due to the vastness of the literature in fictitious play, it is not possible to include all the works that have either been used or been inspired by this method.
lower-left element, the sequence of successive increments form a spiraling trajectory that converge towards the element of maximum value. We demonstrate that fictitious play has to follow the same trajectory, passing through all non-zero elements, to reach the unique pure Nash equilibrium, as illustrated in Figure 2. A crucial component of the proof is provided by an induction argument that emulates the movement of fictitious play and provides a super-exponential lower bound on the number of rounds needed.
2.1 Notation and Definitions
Notation Let  $\mathbb{R}$  be the set of real numbers, and  $[n] = \{1,2,\dots ,n\}$  be the set of actions. We define  $\Delta_{n}$  as the probability simplex, which is the set of  $n$  -dimensional probability vectors, i.e.,  $\Delta_n\coloneqq \{x\in \mathbb{R}^n:x_i\geq 0,\sum_{i = 1}^n x_i = 1\}$ . We use  $e_i$  to denote the  $i$  -th elementary vector, and to refer to a coordinate of a vector, we use either  $x_{i}$  or  $[x]_i$ . The superscripts are used to indicate the time at which a vector is referring to.
Normal-form Games In a two-player normal-form game we are given a pair of payoff matrices  $A \in \mathbb{R}^{n \times m}$  and  $B \in \mathbb{R}^{n \times m}$  where  $n$  and  $m$  are the respective pure strategies of the row and the column player. If the row player selects strategy  $i \in [n]$ , and the column player selects strategy  $j \in [m]$ , then their respective payoffs are  $A_{ij}$  for the row player and  $B_{ij}$  for the column player.
The agents can use randomization. A mixed strategy for the row player is a probability distribution  $x \in \Delta_{n}$  over the  $n$  rows, and a mixed strategy for the column player is a probability distribution  $y \in \Delta_{m}$  over the  $m$  columns. After the row player selects mixed strategy  $x \in \Delta_{n}$ , and the column player selects mixed strategy  $y \in \Delta_{m}$ , their expected payoffs are  $x^{\top}Ay$  and  $x^{\top}By$ , respectively.
Potential Games A two-player potential game is a class of games that admit a unique function  $\Phi$ , referred to as a potential function, which captures the incentive of all players to modify their strategies. In other words, if a player deviates from their strategy, then the difference in payoffs is determined by a potential function  $\Phi$  evaluated at those two strategy profiles. We can express this formally as follows:
Definition 2.1 (Two-player Potential Game). For any given pair of strategies  $(x,y)$  and a pair of unilateral deviations  $x^{\prime}$  by the row player and  $y^\prime$  by the column player, the difference in their utility is equivalent to the difference in the potential function.
$$
\left(x ^ {\prime}\right) ^ {\top} A y - x ^ {\top} A y = \Phi \left(x ^ {\prime}, y\right) - \Phi (x, y) \quad \text {a n d} \quad x ^ {\top} A \left(y ^ {\prime}\right) - x ^ {\top} A y = \Phi \left(x, y ^ {\prime}\right) - \Phi (x, y)
$$
Remark 2.2. We note  $\Phi$  is a function that characterizes the equilibria of the game as the strategy profiles that maximize the potential function.
In this work, we focus on a specific type of potential games called identical payoff games, where both players receive the same payoff.
Definition 2.3 (Identical Payoff Games). A two-player normal-form game  $(A, B)$  is called identical payoff if and only if  $A = B$ .
In this scenario, it is apparent that the potential function is given by  $\Phi (x,y) = x^{\top}Ay$ . Finally we provide the definition of an approximate NE.
Definition 2.4 ( $\epsilon$ -Nash Equilibrium). A strategy profile  $(x^{\star}, y^{\star}) \in \Delta_{n} \times \Delta_{m}$  is called an  $\epsilon$ -approximate NE if and only if
$$
\left(x ^ {\star}\right) ^ {\top} A y ^ {\star} \geq x ^ {\top} A y ^ {\star} - \epsilon \forall x \in \Delta_ {n} \quad \text {a n d} \quad \left(x ^ {\star}\right) ^ {\top} B y ^ {\star} \geq \left(x ^ {\star}\right) ^ {\top} B y - \epsilon \forall y \in \Delta_ {m}
$$
In words, an approximate Nash equilibrium is a strategy profile in which no player can improve their payoff significantly by unilaterally changing their strategy, but the strategy profile may not necessarily satisfy the precise definition of a Nash equilibrium.
Remark 2.5. We highlight two special cases of the  $\epsilon$ -NE. Firstly, when  $\epsilon$  is equal to zero, it is referred to as an exact Nash equilibrium. Secondly, when the support of strategies is of size 1, it is called a pure Nash equilibrium. It is worth noting that potential games always admit a pure NE.
2.2 Fictitious Play
Fictitious play is a natural uncoupled game dynamics at which each agent chooses a best response to their opponent's empirical mixed strategy. Since there might be several best response actions at a given round, fictitious play might contain different sequences of play; see Definition 2.6.
Definition 2.6 (Fictitious Play). An infinite sequence of pure strategy profiles  $(i^{(1)},j^{(1)}),\ldots ,(i^{(t)},j^{(t)})$  ,... is called a fictitious play sequence if and only if at each round  $t\geq 2$
$$
i ^ {(t)} \in \operatorname {a r g m a x} _ {i \in [ n ]} \sum_ {s = 1} ^ {t - 1} A _ {i j ^ {(s)}} \quad \text {a n d} \quad j ^ {(t)} \in \operatorname {a r g m a x} _ {j \in [ m ]} \sum_ {s = 1} ^ {t - 1} B _ {i ^ {(s)} j} \tag {1}
$$
The empirical strategy profile of row and column player at time  $T$  is defined as  $\hat{x}^{(T)} = \left(\frac{1}{T}\sum_{s=1}^{T}e_{i(s)}\right)$  and  $\hat{y}^{(T)} = \left(\frac{1}{T}\sum_{s=1}^{T}e_{j(s)}\right)$  where  $e_{i(t)}, e_{j(t)}$  are the elementary basis vectors.
Definition 2.7 (Cumulative utility vector). For an infinite sequence of pure strategy profiles  $(i^{(1)},j^{(1)}),\ldots ,(i^{(t)},j^{(t)})$  , ., the cumulative utility vectors of the row and column player at round  $t\geq 1$  are defined as,
$$
R ^ {(t)} = \sum_ {s = 1} ^ {t - 1} A e _ {j ^ {(s)}} \quad \text {a n d} \quad C ^ {(t)} = \sum_ {s = 1} ^ {t - 1} e _ {i ^ {(s)}} ^ {\top} B.
$$
Remark 2.8. Fictitious play assumes that each agent selects at each round  $t \in [T]$  a strategy with maximum cumulative utility. The latter decision-making algorithm is also known as Follow the Leader. We remark that the latter alternative interpretation provides a direct generalization of fictitious play in  $N$ -player games.
In their seminal work, Monderer et al.[18] established that in case of identical payoff games the empirical strategies of any fictitious play sequence converges asymptotically to a NE.
Theorem 2.9 ([18]). Let a fictitious play sequence  $(i^{(1)}, j^{(1)}), \ldots, (i^{(t)}, j^{(t)}), \ldots$  for an identical payoff game described with matrix  $A$ . Then, there exists a round  $T^{\star} \geq 1$  such that for any  $t \geq T^{\star}$ , the empirical strategy profile  $(\hat{x}^{(t)}, \hat{y}^{(t)})$  converges to a NE with a rate of  $1/t$ .
On the positive side, Theorem 2.9 establishes that any fictitious play sequence converges to a Nash equilibrium in the case of potential games  $^*$  On the negative side, Theorem 2.9 does not provide any convergence rates, since the round  $T^{\star}$  depends on the specific fictitious play sequence and its dependence on the number of strategies is rather unclear.
In this section, we outline the steps towards proving Theorem 3.1, as it is stated below. Firstly, we introduce a carefully constructed payoff matrix  $A$  of size  $n \times n$  and analyze its structural properties in Section 3.1. Next, in Section 3.2, we investigate the behavior of fictitious play when the game is an two-player identical payoff game with this matrix  $A$ . We also present a set of key statements that are necessary for proving the main theorem. Finally, in Section 3.3, we provide a proof for the fundamental Lemma 3.8.
Theorem 3.1. Let an identical payoff game defined with the matrix  $A$  of size  $n \times n$  and consider any fictitious play sequence  $(i^{(1)}, j^{(1)}), \ldots, (i^{(t)}, j^{(t)}), \ldots$  with  $(i^{(1)}, j^{(1)}) = (n, 1)$ . In case the empirical strategy profile  $(\hat{x}^{(T)}, \hat{y}^{(T)})$  is an  $\epsilon$ -approximate Nash equilibrium then it holds
$$
T \geq \Omega \left(4 ^ {n} ((n / 2 - 2)!) ^ {4} + \frac {1}{n \sqrt {\epsilon}}\right).
$$
Moreover, the lower bound on  $T$  is independent of the tie-breaking rule. Finally, if the initialization is chosen uniformly at random, then the expected number of rounds to reach an  $\epsilon$ -approximate Nash equilibrium is  $\Omega \left(\frac{4^n((n/2-2)!)^4 + 1/n\sqrt{\epsilon}}{n^2}\right)$ .
*For the sake of exposition, we have stated Theorem 2.9 only for the case of identical payoff games. However, we remark that the same theorem holds for general  $N$ -player potential games.
$$
K ^ {n} (z) = \left( \begin{array}{c c c} (z + 2) & 0 & (z + 3) \\ 0 & & 0 \\ \vdots & K ^ {n - 2} (z + 4) & \vdots \\ 0 & & (z + 4) \\ (z + 1) & 0 & 0 \end{array} \right)
$$
(a) Recursive construction of  $A$
$$
K ^ {6} (0) = \left( \begin{array}{c c c c c c} 2 & 0 & 0 & 0 & 0 & 3 \\ 0 & 6 & 0 & 0 & 7 & 0 \\ 0 & 0 & 1 0 & 1 1 & 0 & 0 \\ 0 & 0 & 9 & 0 & 8 & 0 \\ 0 & 5 & 0 & 0 & 0 & 4 \\ 1 & 0 & 0 & 0 & 0 & 0 \end{array} \right)
$$
(b) An example for  $z = 0$  and  $n = 6$
3.1 Construction and Analysis of the Payoff Matrix  $A$
We begin by introducing our recursive construction for the payoff matrix, which we use to establish the formal statement of Theorem 3.1.
Definition 3.2. For any  $z > 0$  and  $n$  even, consider the following  $n \times n$  matrix  $K^n(z)$ .
1. For  $n = 2$ ,  $K^n(z) = \begin{pmatrix} z + 2 & z + 3 \\ z + 1 & 0 \end{pmatrix}$ .
2. For  $n\geq 4$
-  $K_{n1}^{n}(z) = z + 1$  and  $K_{nj}^{n}(z) = 0$  for  $j \notin \{1\}$ .
Row n
-  $K_{n1}^{n}(z) = z + 1$ ,  $K_{11}^{n}(z) = z + 2$  and  $K_{1j}^{n}(z) = 0$  for any  $j \notin \{1, n\}$ .
-  $K_{11}^{n}(z) = z + 2$ ,  $K_{1n}^{n}(z) = z + 3$  and  $K_{1j}^{n}(z) = 0$  for  $j \notin \{1, n\}$ .
Row 1
-  $K_{1n}^{n}(z) = z + 3$ ,  $K_{n - 1n}^{n}(z) = z + 4$  and  $K_{nj}^{n}(z) = 0$  for  $j \notin \{1, n - 1\}$ .
Column n
- For all  $i, j \in \{2, \ldots, n - 1\} \times \{2, \ldots, n - 1\}$ ,  $K_{ij}^{n}(z) \coloneqq K_{i - 1j - 1}^{n - 2}(z + 4)$ .
In Figures 1a and 1b, we provide a schematic representation of Definition 3.2 and an illustrative example for  $n = 6$  and  $z = 0$ . For the sake of simplicity, we have intentionally omitted the remaining zeros in the outer rows, and columns of the matrix.
The construction of the payoff matrix exhibits an interesting circular pattern, which begins at the lower left corner and extends along the outer layer of the matrix. More specifically, the first increment occurs in the same column as the starting point, i.e., at position  $(1,1)$  on the first row. The pattern then proceeds to the next greater element on the same row but a different column, i.e., at position  $(1,n)$ , and the last increment before entering the inner sub-matrix is located on the same column but on the  $(n-1)$ -th row.
$$
\text {S e q u e n c e o f i n c r e m e n t s :} (n, 1) \rightarrow (1, 1) \rightarrow (1, n) \rightarrow (n - 1, n) \rightarrow \underbrace {(n - 1 , 1) \rightarrow \cdots} _ {K ^ {n - 2} (z + 4)}
$$
The increments have been carefully selected to ensure that there are alternating changes in row and column when starting from the lower-left corner and following the successive increments, until reaching the sub-matrix in the center. Once inside the sub-matrix, a similar pattern continues. As we will explore later on, the structure of the payoff dictates the behavior of fictitious play. We denote the payoff matrix under consideration as  $A$ , which is defined as  $A \coloneqq K_n(0)$ . The subsequent statements establish the key properties of  $A$ .
Observation 3.3 (Structural Properties of matrix  $A$ ). Let the matrix  $A = K^n(0)$ , then for  $i \in \{0, \dots, \frac{n}{2} - 1\}$  the following hold:
- The only elements with non-zero values in column  $i + 1$  are located at positions  $i + 1$  and  $n - i$ , and have values  $4i + 2$  and  $4i + 1$ , respectively.
- The only non-zero elements of row  $i + 1$  are located at positions  $i + 1$  and  $n - i$ , and have values  $4i + 2$  and  $4i + 3$ , respectively.
- The only non-zero elements of column  $n - i$  are located at positions  $i + 1$  and  $n - i - 1$  and have values  $4i + 3$  and  $4i + 4$ , respectively.
- The only non-zero elements of row  $n - i$  are located at positions  $i + 1$  and  $n - i + 1$ , and have values  $4i + 1$  and  $4i$ , respectively.
Observation 3.4. The maximum value of  $A$  is  $2n - 2$  and is located at the entry  $\left(\frac{n}{2}, \frac{n}{2} + 1\right)$ .
Proposition 3.5. For any non-zero element in the matrix  $A$ , there is at most one non-zero element that is greater and at most one non-zero element that is smaller in the same column or row.
Proof. By Observation 3.3, each row and column of matrix  $A$  contains at most two non-zero elements that are necessarily different to each other. Thus, if  $(i,j)$  is a non-zero element of  $A$ , there can be at most two additional non-zero elements in row  $i$  and column  $j$  combined. Moreover, at most one of these elements can be greater and at most one can be smaller.
One of the central components of the main theorem is presented below. Lemma 3.6 establishes that in an identical payoff game with matrix  $A$ , any approximate Nash equilibrium must distribute the majority of its probability mass to the maximum element in  $A$ , which is located at the entry  $\left(\frac{n}{2}, \frac{n}{2} + 1\right)$ . The proof of this theorem is based solely on the structural properties presented in Observation 3.3, and is provided in the Appendix.
Lemma 3.6 (Unique  $\epsilon^2$ -NE). Let  $\epsilon \in O\left(\frac{1}{n^3}\right)$  and consider an  $\epsilon^2$ -approximate Nash Equilibrium  $(x^{*},y^{*})$ . Then the following hold,
$$
x _ {\frac {n}{2}} ^ {*} \geq 1 - n \epsilon \quad a n d \quad y _ {\frac {n}{2} + 1} ^ {*} \geq 1 - n \epsilon
$$
Lemma 3.6 not only establishes that the only approximate Nash equilibrium of the identical-payoff game with matrix  $A$  corresponds to the strategies  $\left(\frac{n}{2}, \frac{n}{2} + 1\right)$ , but it also implies that this is the only exact Nash equilibrium (i.e.,  $\epsilon = 0$ ). This observation follows from Observation 3.4, which states that the entry  $\left(\frac{n}{2}, \frac{n}{2} + 1\right)$  corresponds to a maximum value of  $A$  and hence it is a Nash equilibrium as it dominates both the row and the column that it belongs to. We can formally state this observation as follows.
Corollary 3.7 (Unique pure NE). In an identical-payoff game with payoff matrix  $A$ , there exists a unique pure Nash equilibrium at  $\left( \frac{n}{2}, \frac{n}{2} + 1 \right)$ .
3.2 Lower Bound for Fictitious Play in a Game with Matrix  $A$
In this subsection, we present the proof of Theorem 3.1. To achieve this, we first prove that fictitious play requires super-exponential time before placing a positive amount of mass in entry  $\left(\frac{n}{2},\frac{n}{2} +1\right)$ . This result is established by our main technical contribution of the subsection, which is Lemma 3.8.
Lemma 3.8. Let an identical-payoff game with payoff matrix  $A$  and a fictitious play sequence  $(i^{(1)}, j^{(1)}), \ldots, (i^{(t)}, j^{(t)})$ , ... with  $(i^{(1)}, j^{(1)}) = (n, 1)$ . Then, for all  $\ell = \{0, \dots, \frac{n}{2} - 1\}$  there exists a round  $T_{\ell} \geq 1$  such that:
1. the agents play the strategies  $(n - \ell, \ell + 1)$  for the first time,
2. all rows  $r \in [\ell + 1, n - \ell - 1]$  admit 0 cumulative utility,  $R_r^{(T_\ell)} = 0$ ,
3. all columns  $c \in [\ell + 2, n - \ell]$  admit 0 cumulative utility,  $C_c^{(T_\ell)} = 0$ .
Moreover for  $\ell \geq 2$ , the cumulative utility of row  $n - \ell$  at round  $T_{\ell}$  is greater than
$$
R _ {n - \ell} ^ {(T _ {\ell})} \geq 4 \ell (4 \ell - 1) (4 \ell - 2) (4 \ell - 3) \cdot R _ {n - \ell} ^ {(T _ {\ell - 1})} \text {w h i l e} R _ {n - 1} ^ {(T _ {1})} \geq 4. \tag {2}
$$
Using Lemma 3.8 we are able to establish that for a very long period of time the row player has never played row  $\frac{n}{2}$  and that the column player has never played column  $\frac{n}{2} + 1$ .
Lemma 3.9 (Exponential Lower Bound). Let an identical-payoff game with matrix  $A$  and a fictitious play sequence  $(i^{(1)}, j^{(1)}), \ldots, (i^{(t)}, j^{(t)})$ , ... with  $(i^{(1)}, j^{(1)}) = (n, 1)$ . In case  $(i^{(T)}, j^{(T)}) = (\frac{n}{2}, \frac{n}{2} + 1)$  then  $T \geq \Omega (4^n ((n/2 - 2)!)^4)$ .
Figure 2: This figure shows the spiral trajectory generated by the fictitious play dynamic in a game with matrix  $A$ . The starting point is the lower-left element, and as the dynamic progresses, it visits all non-zero elements in ascending order of value.