$11\mathrm{.}8$ Exercises Consider a trading network with intermediaries in which there is one seller S $,$ two buyers B$1,$B$2$ and two traders $($intermediaries$)$ T$1,$T$2.$ The seller is allowed to trade with either trader. The buyers can each trade with only one of the traders: buyer B$1$ can only trade with trader T$1$ ; and buyer B$2$ can only trade with trader T$2.$ The seller has one unit of the object and values it at $0$ ; the buyers are not endowed with the object. Buyer B$1$ values a unit at $1$ and buyer B$2$ values a unit at $2.($a$)$ Draw the trading network, with the traders as squares, the buyers and the seller as circles, and with edges connecting nodes who are able to trade with each other. Label each node as S$,$B$1,$B$2,$T$1$ or T$2.($b$)$ Suppose the traders offer prices aly follows. Trader T$1$ offers a bid price of $(1)/(3)$ to S and an ask price of $1$ to B$1.$ Trader T$2$ offers a bid price of $(2)/(3)$ to S and an ask price of $2$ to B$2.$ Does this set of prices form a Nash equilibrium? If you think the answer is yes, give a brief $(1-3$ sentence$)$ explanation why. If you think the answer is no$,$ describe a way in which one of the traders could changes its prices so as to increase its profit. Consider a trading network with intermediaries in which there are three sellers $S1,S2,S3$

two buyers $B1,B2,$ and two traders $($intermediaries$)T1,T2.$ Sellers $S1$ and $S2$ can

trade only with trader $T1$; and, seller $S3$ can trade only with trader $T2.$ The buyers

can each trade with only one of the traders: buyer $B1$ can only trade with trader $T1$;

and buyer $B2$ can only trade with trader $T2.$ The sellers each have one unit of the

object and value it at $0.$ The buyers are not endowed with the object and they each

value a unit at $1.$

$($a$)$ Draw the trading network, with the traders as squares, the buyers and sellers as

circles, and with edges connecting nodes who are able to trade with each other. Label

each node as $S1,S2,S3,B1,B2,T1$ or $T2.$

$($b$)$ Describe what the possible Nash equilibria are, including both prices and the flow

of goods. Give an explanation for your answer.

$($c$)$ Suppose now that we add an edge between buyer $B2$ and trader $T1.$ We want to

examine whether this new edge changes the outcome in the game. To do this, take

the equilibrium from your answer to $($b$),$ keep the prices and the flow of goods on the

edges from $($b$)$ the same as before, and then suppose that the ask price on the new

$B2-T1$ edge is $1,$ and that no good flows on this new edge. Do these prices and this

overall flow of goods still form an equilibrium? If you think that the answer is yes,

give a brief $(1-3$ sentence$)$ explanation why. If you think the answer is no$,$ describe a

way in which one of the participants in the game would deviate.