Trading in NetworksSlide 2Trade EconomiesCash and PricesMathematical MicroeconomicsMarket EquilibriumExamplesAnother Phone Call from StockholmRemarksNetworked Trade: MotivationNetworked Trade: A ModelNetwork EquilibriumNetwork Structure and OutcomeSlide 14Slide 15A More Complex ExampleCharacterizing Price VariationA Bipartite Network Formation ModelSlide 19(Statistical) Structure and OutcomeSlide 21Slide 22Slide 23Slide 24Slide 25Behavioral Experiments in Networked TradeGame OverviewSlide 28Slide 29Slide 30Slide 31Slide 32Trading in NetworksNetworked LifeCIS 112Spring 2010Prof. Michael Kearnsstrategic gamesNash equilibriumnetworked gamesbehaviortrade economiesprice equilibriumnetworked tradebehaviorTrade Economies•Suppose there are a bunch of different goods orcommodities –wheat, milk, rice, paper, raccoon pelts, matches, grain alcohol,…–commodity = no differences or distinctions within a good: rice is rice•We may all have different initial amounts or endowments–I might have 10 sacks of rice and two raccoon pelts–you might have 6 bushels of wheat, 2 boxes of matches–etc. etc. etc.•Of course, we may want to trade or exchange some of our goods–I can’t eat 10 sacks of rice, and I need matches to light a fire–it’s getting cold and you need raccoon mittens–etc. etc. etc.•How should we engage in trade?•What should be the rates of trade?–how many sacks of rice per box of matches?•These are among the oldest questions in markets and economics•Obviously can be specialized to “modern” markets (e.g. stocks)Cash and Prices•Suppose we introduce an abstract resource called cash–no inherent value–simply meant to facilitate trade; “encode” pairwise exchange rates•And now suppose we introduce prices in cash (from where?)–i.e. rates of exchange between each “real” good and cash–e.g. a raccoon pelt is worth $5.25, a box of matches $1.10•Then if we all believed in cash and the prices…–we might try to sell our initial endowments for cash–then use the cash to buy exactly what we most want•But will there really be:–others who want to buy all of our endowments? (demand)–others who will be selling what we want? (supply)–how might we find them?•A complex, distributed market coordination problemMathematical Microeconomics•Have k abstract goods or commodities g1, g2, … , gk•Have n consumers or “players”•Each player has an initial endowment e = (e1,e2,…,ek) > 0•Each consumer has their own utility function:–assigns a subjective “valuation” or utility to any amounts of the k goods–e.g. if k = 4, U(x1,x2,x3,x4) = 0.2*x1 + 0.7*x2 + 0.3*x3 + 0.5*x4 (* = multiplication)•this is an example of a linear utility function, with weights/coefficients w_i•lots of other possibilities; e.g. diminishing utility as amount becomes large–here g2 is my “favorite” good --- but it might be expensive•Rationality = utility maximization subject to budget–e.g. suppose U as above and prices p = (1.0,0.35,0.15,2.0) per unit–look at “bang for the buck” for each good i, w_i/p_i:•g1: 0.2/1.0 = 0.2; g2: 0.7/0.35 = 2.0; g3: 0.3/0.15 = 2.0; g4: 0.5/2.0 = 0.25•so we will purchase as much of g2 and/or g3 as we can subject to our budget•we are indifferent between g2 and g3Market Equilibrium•Suppose we “announce” prices p = (p1,p2,…,pk) for the k goods•Assume consumers are rational:–they will attempt to sell their endowment e at the posted prices p (supply)–if successful, they will get cash C = e1*p1 + e2*p2 + … + ek*pk (* = times)–with this cash, they will then attempt to purchase x = (x1,x2,…,xk) that maximizes their utility U(x) subject to their budget C (demand), as per the last slide•A specific mechanism: –bring your endowments to the stage–I act as banker, distribute cash for endowments–return to stage, use cash to buy optimal bundle of goods•What could go wrong? –1) stuff left on stage 2) not enough stuff on stage•Say that the prices p are an equilibrium if there are exactly enough goods to accomplish all supply and demand constraints•That is, supply exactly balances demand --- market clearsExamples•Example 1: 3 consumers, 2 goods–Consumer A: utility 0.5*x1 + 0.5*x2 (indifferent)–Consumer B: utility 0.75*x1 + 0.25*x2 (prefers Good 1)–Consumer C: utility 0.25*x1 + 0.75*x2 (prefers Good 2)–all endowments = (1,1)•Claim: equilibrium prices = (1.0,1.0)–all three consumers receive 2.0 from sale of endowments–3 units of Good 1: •Consumer B buys as much as he can  2 units–3 units of Good 2:•Consumer C buys as much as he can  2 units–1 unit remains of each good•Consumer A is indifferent, buys both•Example of non-equilibrium prices: (1.0, 2.0)–all three consumers get 3.0 from endowments–Consumers A and B each want to buy 3 units of Good 1, but there’s not enough–Consumer C can buy 1.5 units of Good 2, but that leaves 1.5 units of surplus–important: consumers don’t have to worry about balancing supply demand --- prices should do that•Example 2:–Consumer A: utility 0.5*x1 + 0.5*x2 (indifferent)–Consumer B: 1.0*x1 (prefers Good 1)–Consumer C: 1.0*x1 (also prefers Good 1)–all endowments = (1,1)•Claim: equilibrium prices = (2.0,1.0)–All three consumers receive 2+1 = 3.0 from sale of endowments–3 units of Good 1:•Consumer B buys as much as he can  1.5 units•Consumer C buys as much as he can  1.5 units•supply of Good 1 is exhausted–3 units of Good 2•Consumer A can exactly purchase all 3•How did I figure this out? Guess that B and C must split Good 1  1.5*p1 = 3*p2•Note: even for centralized computation, finding equilibrium is challenging (but tractable)Another Phone Call from Stockholm•Arrow and Debreu, 1954:–there is always a set of equilibrium prices!–no matter how many consumers & goods, any utility functions, etc.–both won Nobel prizes in Economics•Intuition: suppose p is not an equilibrium–if there is excess demand for some good at p, raise its price–if there is excess supply for some good at p, lower its price–the famed “invisible hand” of the market•The problems with this intuition:–changing prices can radically alter consumer preferences•not necessarily a gradual process; see “bang for the buck” argument–everyone reacting/adjusting simultaneously–utility functions may be extremely complex•May also have to specify “consumption plans”:–who buys exactly what, and