Survey

Survey

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Networked Trade: Theory and Behavior Networked Life CIS 112 Spring 2009 Prof. Michael Kearns strategic games trade economies Nash equilibrium price equilibrium networked games networked trade behavior behavior Trade Economies • Suppose there are a bunch of different goods orcommodities • We may all have different initial amounts or endowments • Of course, we may want to trade or exchange some of our goods • • How should we engage in trade? What should be the rates of trade? • • These are among the oldest questions in markets and economics Obviously can be specialized to “modern” markets (e.g. stocks) – wheat, milk, rice, paper, raccoon pelts, matches, grain alcohol,… – commodity = no differences or distinctions within a good: rice is rice – 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. – 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 many sacks of rice per box of matches? Cash and Prices • Suppose we introduce an abstract resource called cash • And now suppose we introduce prices in cash (from where?) • Then if we all believed in cash and the prices… • But will there really be: • A complex, distributed market coordination problem – no inherent value – simply meant to facilitate trade; “encode” pairwise exchange rates – 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 – we might try to sell our initial endowments for cash – then use the cash to buy exactly what we most want – others who want to buy all of our endowments? (demand) – others who will be selling what we want? (supply) – how might we find them? Mathematical 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 • lots of other possibilities; e.g. diminishing utility as amount becomes large – here g2 is my “favorite” good --- but it might be expensive – generally assume utility functions are insatiable • always some bundle of goods you’d prefer more Market 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 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) – example: • U(x1,x2,x3,x4) = 0.2*x1 + 0.7*x2 + 0.3*x3 + 0.5*x4 • p = (1.0,0.35,0.15,2.0) • look at “bang for the buck” for each good i, wi/pi: – – 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 budget • A specific mechanism: • What could go wrong? • 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 clears • – 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 – 1) stuff left on stage 2) not enough stuff on stage Examples • Example 1: 3 consumers, 2 goods • Claim: equilibrium prices = (1.0,1.0) – – – – 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) – – all three consumers receive 2.0 from sale of endowments 3 units of Good 1: – 3 units of Good 2: – 1 unit remains of each good • Consumer B buys as much as he can 2 units • Consumer C buys as much as he can 2 units • Consumer A is indifferent, buys both • Example 2: • Claim: equilibrium prices = (2.0,1.0) • • – – – – 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) – – All three consumers receive 2+1 = 3.0 from sale of endowments 3 units of Good 1: – 3 units of Good 2 • • • 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 • 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 = p1+p2 Note: even for centralized computation, finding equilibrium is challenging (but tractable) Another Phone Call from Stockholm • Arrow and Debreu, 1954: • Intuition: suppose p is not an equilibrium • The problems with this intuition: – there is always a set of equilibrium prices! – no matter how many consumers & goods, any utility functions, etc. – both won Nobel prizes in Economics – 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 – 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 from whom – in previous example, may have to specify how much of g2 and g3 to buy – example: • A has Fruit Loops and Lucky Charms, but wants granola • B and C have only granola, both want either FL or LC (indifferent) • need to “coordinate” B and C to buy A’s FL and LC Remarks • A&D 1954 a mathematical tour-de-force • Actual markets have been around for millennia • Model abstracts away details of price adjustment/formation process • Model can be augmented in various way: • “Efficient markets” ~ in equilibrium (at least at any given moment) – resolved and clarified a hundred of years of confusion – proof related to Nash’s; (n+1)-player game with “price player” – highly structured social systems – it’s the mathematical formalism and understanding that’s new – – – – – does not specify any particular “mechanism” modern financial markets pre-currency bartering and trade auctions etc. etc. etc. – labor as a commodity – firms producing goods from raw materials and labor – etc. etc. etc. Networked Trade: Motivation • All of what we’ve said so far assumes: • But there are many economic settings in which everyone is not free to directly trade with everyone else – that anyone can trade (buy or sell) with anyone else – equivalently, exchange takes place on a complete network – at equilibrium, global prices must emerge due to competition – geography: • perishability: you buy groceries from local markets so it won’t spoil • labor: you purchases services from local residents – legality: • if one were to purchase drugs, it is likely to be from an acquaintance (no centralized market possible) • peer-to-peer music exchange – politics: • there may be trade embargoes between nations – regulations: • • on Wall Street, certain transactions (within a firm) may be prohibited Nice real-world example of a market with strong network constraints: electricity markets • e.g. PJM Interconnect • challenges of electricity storage, regional generation & consumption Networked Trade: A Model • Still begin with the same framework: • But now assume an undirected network dictating exchange • Note: can “encode” network in goods and utilities – k goods or commodities – n consumers, each with their own endowments and utility functions – – – – each vertex represents a consumer edge between i and j means they are free to engage in trade no edge between i and j: direct trade is forbidden simplest case: no “resale” allowed --- one “round” of trading – for each raw good g and consumer i, introduce virtual good (g,i) – think of (g,i) as “good g when sold by consumer i” – consumer j will have • zero utility for (g,i) if no edge between i and j • j’s original utility for g if there is an edge between i and j Network Equilibrium • Now prices are for each (g,i), not for just raw goods • Each consumer must still behave rationally • Market equilibrium still always exists! – permits the possibility of variation in price for raw goods – prices of (g,i) and (g,j) may differ – Q: What would cause such variation at equilibrium? – attempt to sell all of initial endowment --- but only to NW neighbors – attempt to purchase goods maximizing utility within budget --- from neighbors – will only purchase g from those neighbors with minimum price for g – set of prices (and consumptions plans) such that: • all initial endowments sold (no excess supply) • no consumer has money left over (no excess demand) • no trades except between network neighbors! Network Structure and Outcome • • • • Q: How does the structure of a network influence the prices/wealths at equilibrium? Need to separate asymmetries of endowments & utilities from those of NW structure We will thus consider bipartite economies Only two kinds of players/consumers: • • • Equal numbers of Milks and Wheats Network is bipartite --- only have edges between Milks and Wheats When will such a network have variation in prices? – – – “Milks”: start with 1 unit of milk, but have utility only for wheat “Wheats” start with 1 unit of wheat, but have utility only for milk exact form of utility functions irrelevant An Example • 2 a 2/3 b 2/3 c 2/3 d w x y z 1/2 3/2 1/2 3/2 • • • Price = amount of the other good received = wealth Prices at opposite ends of any used edge always reciprocal: p and 1/p Checking equilibrium conditions: – only “cheapest” edges used – supply and demand balance: • • • • • • • • a sends 1/2 each to w and y b sends 1 to x c sends 1/2 each to x and z d sends 1 to z w sends 1 to a x sends 2/3 to b, 1/3 to c y sends 1 to a z sends 1/3 to c, 2/3 to d Some edges unused at equilibrium – exchange subgraph 1 a 1 b 1 c 1 d w x y z 1 1 1 1 • • Suppose we add the single green edge Now equilibrium has no wealth variation! A More Complex Example • Solid edges: – exchange at equilibrium • Dashed edges: – competitive but unused • Dotted edges: – non-competitive prices • Note price variation – 0.33 to 2.00 • Degree alone does not determine price! – e.g. B2 vs. B11 – e.g. S5 vs. S14 Characterizing Price Variation • Consider any bipartite “Milk-Wheat” network economy • Necessary and sufficient condition for all equilibrium prices and wealths to be equal: • • • • • What if there is no perfect matching subgraph? How large can the price variation be? For any set of vertices S on one side (e.g. Milks), let N(S) be its set of neighbors on the other side Find the S such that |S|/|N(S)| = p is maximized (here |S| is the number of vertices in S) Then the largest price/wealth in the network will be p, and the smallest 1/p Intuition: When S is very large but N(S) is small, consumers in S are “captives” of their neighbors N(S) • • • Note: When network has a perfect matching, N(S) is always at least as large as S Note: Finding the maximizing set S may involve some computation… Now let’s examine price variation in a statistical network formation model… – again, all endowments equal to 1.0, equal numbers of Milks and Wheats – – network has a perfect matching as a subgraph a pairing of Milks and Wheats such that everyone has exactly one trading partner on the other side – Can actually iterate: remove S and N(S) from the network, find S’ maximizing |S’|/|N(S’)|,… A Bipartite Economy Network Formation Model • • Consider economies with only two goods: milk and wheat… …and only two kinds of players/consumers: • • • • • Wheats and Milks added incrementally in pairs at each time step Goal: bipartite network formation model interpolating between P.A. and E-R Probabilistically generates a bipartite graph All edges between buyers and sellers Each new party will have n > 1 links back to extant graph • Distribution of new buyer’s links: • So (a,n) characterizes distribution of generative model – – – Milks: start with 1 unit of milk, have utility only for wheat Wheats: start with 1 unit of wheat, have utility only for milk exact form of utility functions irrelevant – – note: n = 1 generates bipartite trees larger n generates cyclical graphs – – – with prob. 1 – a: extant seller chosen w.r.t. preferential attachment with prob. a: extant seller chosen uniformly at random a = 0 is pure pref. att.; a = 1 is “like” Erdos-Renyi model Price Variation vs. a and n n=1 n = 250, scatter plot n=2 Exponential decrease with a; rapid decrease with n (Statistical) Structure and Outcome • Wealth distribution at equilibrium: – Power law (heavy-tailed) in networks generated by preferential attachment – Sharply peaked (Poisson) in random graphs • Price variation (max/min) at equilibrium: – Grows as a root of n in preferential attachment – None in random graphs • Random graphs result in more “socialist” outcomes – Despite lack of centralized formation process An Amusing Case Study U.N. Comtrade Data Network Full Network wealth sorted equilibrium wealth vertex degree USA: 4.42 Germany: 4.01 Italy: 3.67 France: 3.16 Japan: 2.27 European Union Network Full Network EU network price sorted equilibrium prices vertex degree USA: 4.42 Germany: 4.01 Italy: 3.67 France: 3.16 Japan: 2.27 EU: 7.18 USA: 4.50 Japan: 2.96 Behavioral Experiments in Networked Trade Game Overview • • • • Simplified version of classic exchange economies (Arrow-Debreu) Players divided into two equal populations; all graphs bipartite Start with 10 divisible units endowment of either “Milk” or “Wheat” Only value the other good – • Exchange mechanism: – – – – – • payoffs proportional to amount obtained (10 units = $2) can only trade with network neighbors simple limit orders (e.g. offer 2 units Milk for 3 units Wheat) no price discrimination in a neighborhood: prices on vertices, not edges partial executions possible no resale Only source of asymmetry is network position Equilibrium Theory and Network Structure • Equilibrium: set of prices (exchange rates) at which market clears – – – – – • no local supply/demand imbalances accompanied by exchange subgraph; only trade with neighbors offering best prices a static notion; does not specify a trading mechanism network structure may give rise to different prices and wealths throughout the graph centralized computation uses linear programming as a subroutine Theorem: [Kakade, K., Ortiz, Pemantle, Suri] – No wealth variation at equilibrium network contains a perfect matching – Max/min wealth correspond to maximum contraction : large set with few neighbors – degree alone does not determine wealth • • Preferential attachment: wealth imbalance grows with network size Random (Erdos-Renyi) networks: no wealth variation Pairs (1 trial) 2-Cycle (3) 4-Cycle (3) Clan (3) Clan + 5% (3 samples) Clan + 10% (3) demo Erdos-Renyi, p=0.2 (3) E-R, p=0.4 (3) Pref. Att. Tree (3) Pref. Att. Dense (3) Collective Performance and Topology overall mean ~ 0.88 • overall behavioral performance is strong • topology matters; many (but not all) pairs distinguished Equilibrium and Collective Performance correlation ~ -0.8 (p < 0.001) correlation ~ 0.96 (p < 0.001) • greater equilibrium variation behavioral performance degrades • greater equilibrium variation greater behavioral variation Equilibrium and Collective Performance • equilibrium theory relevant: beats degree, uniform, centrality • but best model (so far) tilts towards equality • “network inequality aversion” Behavioral Dynamics: Prices and Volumes mean in first 30s ~ 1.05; last 90s ~ 1.71 (highly sig.) • preponderance of early 1-for-1 trading • may contribute largely to inequality aversion • no rush of trading at the closing Fragmentation of Liquidity Conditional Equilibrium Wealth (CEW): actual earnings so far + equilibrium wealth given (global) trades so far Almost all topology pairs are distinguished by individual CEW variation Cumulative CEW: decreases are structural “traumas” that isolate goods [demo]