# 经济代写|博弈论代写Game Theory代考|ECON40010

#### Doug I. Jones

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## 经济代写|博弈论代写Game Theory代考|Equilibria

When we talk about an “equilibrium” of an utility measure $U \in$ $\mathbb{R}^{\mathfrak{S} \times \mathfrak{S}}$ on the system $\mathfrak{S}$, we make the prior assumption that each state $\sigma$ has associated a neighborhood
$$\overline{\mathcal{F}}^\sigma \subseteq \mathfrak{S} \text { with } \sigma \in \mathcal{F}^\sigma$$
and that we concentrate on state transitions to neighbors, i.e., to transitions of type $\sigma \rightarrow \tau$ with $\tau \in \mathcal{F}^\sigma$.

We now say that a system state $\sigma \in \mathfrak{S}$ is a gain equilibrium of $U$ if no feasible transition $\sigma \rightarrow \tau$ to a neighbor state $\tau$ has a positive utility, i.e., if
$$U(\sigma, \tau) \leq 0 \quad \text { holds for all } \tau \in \mathcal{F}^\sigma .$$
Similarly, $\sigma$ is a cost equilibrium if
$$U(\sigma, \tau) \geq 0 \text { holds for all } \tau \in \mathcal{F}^\sigma .$$
REMARK $5.1$ (Gains and costs). The negative $C=-U$ of the utility measure $U$ is also a utility measure and one finds:
$\sigma$ is a gain equilibrium of $U \Longleftrightarrow \sigma$ is a cost equilibrium of $C$

From an abstract point of view, the theory of gain equilibria is equivalent to the theory of cost equilibria.

Many real-world systems appear to evolve in dynamic processes that eventually settle in an equilibrium state (or at least approximate an equilibrium) according to some utility measure. This phenomenon is strikingly observed in physics. But also economic theory has long suspected that economic systems may tend towards equilibrium states. ${ }^2$

## 经济代写|博弈论代写Game Theory代考|Existence of equilibria

In practice, the determination of an equilibrium is typically a very difficult computational task. Moreover, many utilities do not even admit equilibria. It is generally not easy just to find out whether an equilibrium for a given utility exists at all. Therefore, one is interested in manageable conditions that allow one to conclude that at least one equilibrium exists.

Utilities from potentials. Consider a utility potential $u: \mathfrak{S} \rightarrow \mathbb{R}$ with the marginal utility measure
$$\partial u(\sigma, \tau)=u(\tau)-u(\sigma) .$$
Here, the following sufficient conditions offer themselves immediately:
(1) If $u(\sigma)=\max {\tau \in \mathfrak{S}} u(\tau)$, then $\sigma$ is a gain equilibrium. (2) If $u(\sigma)=\min {\tau \in \mathbb{S}} u(\tau)$, then $\sigma$ is a cost equilibrium.
Since every function on a finite set attains a maximum and a minimum, we find
Proposition 5.2. If $\mathfrak{S}$ is finite, then every utility potential yields a utility measure with at least one gain and one cost equilibrium.

Similarly, we can derive the existence of equilibria in systems that are represented in a coordinate space.
Proposition 5.3. If $\mathfrak{S}$ can be represented as a compact set $\mathcal{S} \subseteq \mathbb{R}^m$ such that $u: \mathcal{S} \rightarrow \mathbb{R}$ is a continuous potential, then $u$ yields a utility measure $\partial u$ with at least one gain and one cost equilibrium.
Indeed, it is well-known that a continuous function on a compact set attains a maximum and a minimum.

REMARK 5.2. Notice that the conditions given in this section are sufficient to guarantee the existence of equilibria – no matter what neighborhood structure on $\mathfrak{S}$ is assumed.

Convex and concave utilities. If the utility measure $U$ under consideration is not implied by a potential function, not even the finiteness of $\mathfrak{S}$ may be a guarantee for the existence of an equilibrium (see Ex. 5.2).

# 博弈论代考

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