2017-07-12 13:46:33 +00:00
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pragma solidity ^0.4.11;
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import "../Utils/Math.sol";
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import "../MarketMakers/MarketMaker.sol";
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/// @title LMSR market maker contract - Calculates share prices based on share distribution and initial funding
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/// @author Alan Lu - <alan.lu@gnosis.pm>
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contract LMSRMarketMaker is MarketMaker {
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using Math for *;
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/*
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* Constants
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*/
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uint constant ONE = 0x10000000000000000;
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int constant EXP_LIMIT = 2352680790717288641401;
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/*
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* Public functions
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*/
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/// @dev Returns cost to buy given number of outcome tokens
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/// @param market Market contract
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/// @param outcomeTokenIndex Index of outcome to buy
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/// @param outcomeTokenCount Number of outcome tokens to buy
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/// @return Cost
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function calcCost(Market market, uint8 outcomeTokenIndex, uint outcomeTokenCount)
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public
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2018-07-02 09:14:28 +00:00
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view
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2017-07-12 13:46:33 +00:00
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returns (uint cost)
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{
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require(market.eventContract().getOutcomeCount() > 1);
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int[] memory netOutcomeTokensSold = getNetOutcomeTokensSold(market);
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// Calculate cost level based on net outcome token balances
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int logN = Math.ln(netOutcomeTokensSold.length * ONE);
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uint funding = market.funding();
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int costLevelBefore = calcCostLevel(logN, netOutcomeTokensSold, funding);
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// Add outcome token count to net outcome token balance
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require(int(outcomeTokenCount) >= 0);
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netOutcomeTokensSold[outcomeTokenIndex] = netOutcomeTokensSold[outcomeTokenIndex].add(int(outcomeTokenCount));
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// Calculate cost level after balance was updated
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int costLevelAfter = calcCostLevel(logN, netOutcomeTokensSold, funding);
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// Calculate cost as cost level difference
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require(costLevelAfter >= costLevelBefore);
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cost = uint(costLevelAfter - costLevelBefore);
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// Take the ceiling to account for rounding
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if (cost / ONE * ONE == cost)
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cost /= ONE;
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else
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// Integer division by ONE ensures there is room to (+ 1)
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cost = cost / ONE + 1;
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// Make sure cost is not bigger than 1 per share
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if (cost > outcomeTokenCount)
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cost = outcomeTokenCount;
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}
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/// @dev Returns profit for selling given number of outcome tokens
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/// @param market Market contract
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/// @param outcomeTokenIndex Index of outcome to sell
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/// @param outcomeTokenCount Number of outcome tokens to sell
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/// @return Profit
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function calcProfit(Market market, uint8 outcomeTokenIndex, uint outcomeTokenCount)
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public
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2018-07-02 09:14:28 +00:00
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view
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2017-07-12 13:46:33 +00:00
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returns (uint profit)
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{
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require(market.eventContract().getOutcomeCount() > 1);
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int[] memory netOutcomeTokensSold = getNetOutcomeTokensSold(market);
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// Calculate cost level based on net outcome token balances
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int logN = Math.ln(netOutcomeTokensSold.length * ONE);
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uint funding = market.funding();
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int costLevelBefore = calcCostLevel(logN, netOutcomeTokensSold, funding);
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// Subtract outcome token count from the net outcome token balance
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require(int(outcomeTokenCount) >= 0);
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netOutcomeTokensSold[outcomeTokenIndex] = netOutcomeTokensSold[outcomeTokenIndex].sub(int(outcomeTokenCount));
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// Calculate cost level after balance was updated
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int costLevelAfter = calcCostLevel(logN, netOutcomeTokensSold, funding);
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// Calculate profit as cost level difference
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require(costLevelBefore >= costLevelAfter);
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// Take the floor
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profit = uint(costLevelBefore - costLevelAfter) / ONE;
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}
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/// @dev Returns marginal price of an outcome
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/// @param market Market contract
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/// @param outcomeTokenIndex Index of outcome to determine marginal price of
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/// @return Marginal price of an outcome as a fixed point number
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function calcMarginalPrice(Market market, uint8 outcomeTokenIndex)
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public
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2018-07-02 09:14:28 +00:00
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view
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2017-07-12 13:46:33 +00:00
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returns (uint price)
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{
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require(market.eventContract().getOutcomeCount() > 1);
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int[] memory netOutcomeTokensSold = getNetOutcomeTokensSold(market);
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int logN = Math.ln(netOutcomeTokensSold.length * ONE);
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uint funding = market.funding();
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// The price function is exp(quantities[i]/b) / sum(exp(q/b) for q in quantities)
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// To avoid overflow, calculate with
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// exp(quantities[i]/b - offset) / sum(exp(q/b - offset) for q in quantities)
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2018-06-21 11:58:38 +00:00
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(uint256 sum, , uint256 outcomeExpTerm) = sumExpOffset(logN, netOutcomeTokensSold, funding, outcomeTokenIndex);
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2017-07-12 13:46:33 +00:00
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return outcomeExpTerm / (sum / ONE);
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}
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/*
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* Private functions
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*/
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/// @dev Calculates the result of the LMSR cost function which is used to
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/// derive prices from the market state
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/// @param logN Logarithm of the number of outcomes
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/// @param netOutcomeTokensSold Net outcome tokens sold by market
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/// @param funding Initial funding for market
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/// @return Cost level
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2018-07-14 21:42:01 +00:00
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function calcCostLevel(int logN, int[] memory netOutcomeTokensSold, uint funding)
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2017-07-12 13:46:33 +00:00
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private
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2018-07-02 09:14:28 +00:00
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view
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2017-07-12 13:46:33 +00:00
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returns(int costLevel)
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{
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// The cost function is C = b * log(sum(exp(q/b) for q in quantities)).
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// To avoid overflow, we need to calc with an exponent offset:
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// C = b * (offset + log(sum(exp(q/b - offset) for q in quantities)))
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2018-06-21 11:58:38 +00:00
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(uint256 sum, int256 offset, ) = sumExpOffset(logN, netOutcomeTokensSold, funding, 0);
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2017-07-12 13:46:33 +00:00
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costLevel = Math.ln(sum);
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costLevel = costLevel.add(offset);
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costLevel = (costLevel.mul(int(ONE)) / logN).mul(int(funding));
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}
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/// @dev Calculates sum(exp(q/b - offset) for q in quantities), where offset is set
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/// so that the sum fits in 248-256 bits
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/// @param logN Logarithm of the number of outcomes
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/// @param netOutcomeTokensSold Net outcome tokens sold by market
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/// @param funding Initial funding for market
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/// @param outcomeIndex Index of exponential term to extract (for use by marginal price function)
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/// @return A result structure composed of the sum, the offset used, and the summand associated with the supplied index
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2018-07-14 21:42:01 +00:00
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function sumExpOffset(int logN, int[] memory netOutcomeTokensSold, uint funding, uint8 outcomeIndex)
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2017-07-12 13:46:33 +00:00
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private
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2018-07-02 09:14:28 +00:00
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view
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2017-07-12 13:46:33 +00:00
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returns (uint sum, int offset, uint outcomeExpTerm)
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{
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// Naive calculation of this causes an overflow
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// since anything above a bit over 133*ONE supplied to exp will explode
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// as exp(133) just about fits into 192 bits of whole number data.
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// The choice of this offset is subject to another limit:
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// computing the inner sum successfully.
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// Since the index is 8 bits, there has to be 8 bits of headroom for
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// each summand, meaning q/b - offset <= exponential_limit,
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// where that limit can be found with `mp.floor(mp.log((2**248 - 1) / ONE) * ONE)`
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// That is what EXP_LIMIT is set to: it is about 127.5
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// finally, if the distribution looks like [BIG, tiny, tiny...], using a
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// BIG offset will cause the tiny quantities to go really negative
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// causing the associated exponentials to vanish.
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int maxQuantity = Math.max(netOutcomeTokensSold);
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require(logN >= 0 && int(funding) >= 0);
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offset = maxQuantity.mul(logN) / int(funding);
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offset = offset.sub(EXP_LIMIT);
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uint term;
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for (uint8 i = 0; i < netOutcomeTokensSold.length; i++) {
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term = Math.exp((netOutcomeTokensSold[i].mul(logN) / int(funding)).sub(offset));
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if (i == outcomeIndex)
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outcomeExpTerm = term;
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sum = sum.add(term);
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}
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}
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/// @dev Gets net outcome tokens sold by market. Since all sets of outcome tokens are backed by
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/// corresponding collateral tokens, the net quantity of a token sold by the market is the
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/// number of collateral tokens (which is the same as the number of outcome tokens the
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/// market created) subtracted by the quantity of that token held by the market.
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/// @param market Market contract
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/// @return Net outcome tokens sold by market
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function getNetOutcomeTokensSold(Market market)
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private
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2018-07-02 09:14:28 +00:00
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view
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2018-07-14 21:42:01 +00:00
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returns (int[] memory quantities)
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2017-07-12 13:46:33 +00:00
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{
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quantities = new int[](market.eventContract().getOutcomeCount());
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for (uint8 i = 0; i < quantities.length; i++)
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quantities[i] = market.netOutcomeTokensSold(i);
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}
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}
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