# Table 1 List of symbols

Symbol Definition
$$\mathcal {C}$$ Campaign set
$$k \in \mathcal {C}$$ A campaign
B k Budget of campaign k in number ofclicks
S k Starting time of the campaign
L k Lifetime of the campaign
$$\mathcal {G}$$ Set of possible user profiles
$$P_{\mathcal {G}}: \mathcal {G} \rightarrow [0,1]$$ Probability that a user belongs to auser profile i
$$i \in \mathcal {G}$$ A user profile
CTR(i,k) Click-through rate indicates theprobability that user i clicks on ad k
c c k Revenue generated by a click oncampaign k
$$\mathcal {S}$$ Set of fully observable MDP states
$$s \in \mathcal {S}$$ A state of the MDP model
$$\mathcal {A}$$ Set of possible actions
$$a \in \mathcal {A}$$ An action of the MDP model
$$\mathcal {A}(s,t)$$ Set of valid actions at instant t in state$$s \in \mathcal {S}$$
$$\mathcal {D}$$ Set of decision epochs
$$t \in \mathcal {D}$$ A decision epoch
$$\mathcal {T}:\mathcal {S}\times \mathcal {A}\times \mathcal {S}\times \mathcal {D}\rightarrow [0,1]$$ Transition function
$$\mathcal {T}(s, a, s', t)$$ Transition from state s to s whenexecutes action a in time t
Reward function
$$\pi :\mathcal {S}\times \mathcal {D} \rightarrow \mathcal {A}$$ Non-stationary deterministic policy
V π(s,t) Value function of policy π
V (s,t) Value function of optimal policy π
$$G \in \mathcal {G} \cup \{0\}$$ User profile that is generating a request(0 stands for no request to attend to)
$$\mathcal {J}_{j}$$ Set of intervals defined by thecampaign time constraints
$$\mathbb {T}_{j}$$ Length of the interval j
x j,i,k Variable that indicates how many adsfrom campaign k should be displayedto users with user profile i at theinterval j
η k Average cost per impression ofcampaign k
ctr k CTR(1,k) for one-profile scenario
R Expected revenue when the AdNetwork applies the solution of theLP relaxation
γ Variable that artificially increases thebudget, $$B_{k}^{'} = \gamma B_{k}$$, γ>1