Consider the class of congestion games discussed in class. here, we focus on the class of parallel congestion games defined as follows a finite set of resources roads r .a congestion function for each resource r of the form er [0,1, -r. the cost cr(k) is the congestion on resource/road r when there are k users a finite set of players n-(1,2 n a finite action set of each player a-r. here, each player can select any single resource r e r. let a : = a, x × an represent the set of joint actions . a cost function for each player i of the form j: a -r that each player seeks to minimize. the specific form of the cost function is where a represents the number of player that chose resource r in the action profile a, 1.e. system cost of the form ien rer (a) is the above congestion game a potential game with the potential function c(a)? (b) is the above congestion game a potential game with the potential function rerk1 (c) prove that a pure nash equilibrium must exist in any parallel congestion game. (d) derive an anonymous tolling scheme, i. e., tr : {1, 2, } → r for each resource r e r such that the new congestion game associated with the player's cost functions is a potential game with potential function c()

Alandowner entered into a written agreement with a real estate broker whereby the broker would receive a commission of 10% of the sale price if he procured a "ready, willing, and able buyer" for the landowner's property and if the sale actually proceeded through closing. the broker found a buyer who agreed in writing to buy the property from the landowner for $100,000, the landowner's asking price. the buyer put up $6,000 as a down payment. the agreement between the landowner and the buyer contained a liquidated damages clause providing that, if the buyer defaulted by failing to tender the balance due of $94,000 at the closing date, damages would be 10% of the purchase price. the landowner included that clause because she was counting on using the proceeds of the sale for a business venture that would likely net her at least $10,000. the buyer became seriously ill and defaulted. when he recovered, he demanded that the landowner return his $6,000, and the landowner refused. the broker also demanded the $6,000 from the landowner and was refused. the broker and the buyer filed separate suits against the landowner, with the buyer pleading impossibility of performance. the two cases are consolidated into a single case. how should the court rule as to the disposition of the $6,000?

Martha is the head of the accounts department in a small manufacturing company. the company follows the accrual-basis method of accounting. it recently purchased raw materials worth $5,000 from its vendors. however, the company paid only $3,000 to its vendors. it plans to pay the remaining amount after three months. considering this information, which entry should martha record in the company’s accounts? a. $5,000 as accounts receivable b. $3,000 as accounts payable c. $2,000 as accounts payable d. $2,000 as accounts receivable

Aprice ceiling is “binding” if the price ceiling is set below the equilibrium price. suppose that the equilibrium price is $5. if a price ceiling is set at $6, this will not affect the market in any way since $5 remains a legally allowable price (since $5 < $6). a price ceiling of $6 is called a “non-binding” price ceiling. on the other hand, if the price ceiling is set at $4, the price ceiling is “binding” because the natural equilibrium price is $5 but that is no longer allowed. what happens when there is a binding price ceiling? at a price below the equilibrium price, quantity demanded exceeds quantity supplied. there is a shortage. normally, price increases eliminate shortages by increasing quantity supplied and decreasing quantity demanded. in this case, however, price increases are not allowed past the price ceiling. we therefore predict that the observed market price will be right at the price ceiling and there will be a permanent shortage. the observed quantity bought and sold will be dictated by the quantity supplied at the price ceiling. although consumers would like to buy more, there are no more units for sale