Poisson Process: Definition, Properties, and Theorems

The Poisson Process

• A counting process, ${N(t): t \geq 0}$, with rate $\lambda > 0$.
• $N(0) = 0$.
• It has independent increments.
• The number of events in any interval of length $t$ follows a Poisson distribution with mean $\lambda t$.

Poisson Distribution Formula

• For all $s, t \geq 0$:
  • $P(N(t+s) - N(s) = n) = e^{-\lambda t} \frac{(\lambda t)^n}{n!}$, for $n = 0, 1, \dots$

Poisson Process Properties

• If ${N(t): t \geq 0}$ is a Poisson process with rate $\lambda$:
  • $E[N(t)] = \lambda t$
  • $Var[N(t)] = \lambda t$

Interarrival Times

• Interarrival times are independent and identically distributed exponential random variables with parameter $\lambda$.
• If the interarrival times of a counting process are independent and identically distributed exponential random variables with parameter $\lambda$, then the counting process is a Poisson process with rate $\lambda$.

Splitting Theorem

• Events of a Poisson process (rate $\lambda$) are classified as either type 1 (probability $p$) or type 2 (probability $1-p$), independently.
• $N_1(t)$ and $N_2(t)$ represent the number of type 1 and type 2 events in $[0, t]$, respectively.
• $N_1(t)$ and $N_2(t)$ are independent Poisson processes with rates $\lambda p$ and $\lambda(1-p)$, respectively.

Algorithmic Game Theory

• The study of mathematical models of strategic interactions among rational agents.

Game Theory Applications

• Found in social science, logic, systems science, and computer science.
• It originally addressed zero-sum games but now applies to a wide range of behavioral relations for decision making in humans, animals, and computers.

Non-Cooperative Game Definition

• A finite set of players $N = {1, 2,..., n}$
• For each player $i \in N$, a finite set of possible strategies $S_i = {s_{i1}, s_{i2},..., s_{ik}}$
• For each player $i \in N$, a utility function $u_i : S_1 \times S_2 \times... \times S_n \rightarrow \mathbb{R}$

Utility Function

• $u_i$ represents the payoff or benefit that player $i$ receives when the players choose the strategies $s_1 \in S_1, s_2 \in S_2,..., s_n \in S_n$.
• The utility function can be written as $u_i(s_1, s_2,..., s_n)$.

Prisoner's Dilemma

• Two suspects are arrested and offered a deal:
  • Confess and the other doesn't: Confessor goes free, the other gets 10 years.
  • Both confess: Each gets 5 years.
  • Neither confesses: Each gets 1 year.
• The utility function:

Nash Equilibrium Definition

• A set of strategies, one for each player, where no player has an incentive to unilaterally change their strategy.
• $(s_1^, s_2^,..., s_n^*)$ such that for every player $i \in N$ and strategy $s_i \in S_i$:
  • $u_i(s_1^, s_2^,..., s_i^,..., s_n^) \geq u_i(s_1^, s_2^,..., s_i,..., s_n^*)$

Prisoner's Dilemma Nash Equilibrium

• Both players confess, because if either player knows the other will confess, their best option is to confess as well.