5. (3 pts) An auction-style exchange has two types of trading orders: standing (limit) and aggressive (market) orders. The investors providing limit orders specify whether they want to sell or to buy, and the price at which they want to trade, and wait until one of the market orders executes their limit order. Of course, each market order can be of a \"sell\" or a \"buy\" type: a market sell order executes the standing buy orders, and a market buy order executes the standing sell orders. A popular predictor of the type of the next market order is the so-called \"market imbalance": a ratio of the total size of all standing buy orders to the total size of all standing sell orders. The higher is market imbalance, the more likely it is that the next market order is of a \"buy\" type. Let time be measured in milliseconds, and let us focus on market buy orders only. As- sume that the market imbalance can be in any of the three states: high (H), medium (M), or low (L). The imbalance may change in either direction for different reasons: because of an incoming market order, or due the cancelation or addition of the stand- ing orders. If the imbalance is low, at the next time step it can either remain the same or increase to medium both with probability 1/2 and no market buy orders can arrive at this time step. If the imbalance is medium, at the next time step it can ei- ther decrease to low, with probability 1K3, or increase to high. When the imbalance increases from medium to high, it may happen for two different reasons: either due to the new market buy order that arrived within this time step, which happens with probability 1f2, or without any new market buy order arriving, which happens with probability 1/6. If the imbalance decreases from medium to low, no market buy or- ders can arrive within this time step. Finally, if the imbalance is high, at the next time step it either decreases to medium or stays at high - both with probability 1/2 and, independent of this, a market order arrives with probability 2 f 3. Consider Betty who always keeps a unit standing sell order in the exchange. Assume that every market buy order will execute Betty's order (after which she will post another one). Every execution gives Betty 1 cent (e.g. from commission or from clever re-purchasing). What is Betty's long-run prot per unit of time? Hint: you need to construct the appmpriarc Markov chain, whose state space shoaid include more ihanjas: H, M and L