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