# How Likely is a Tie in the U.S. Electoral College Vote?

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2 Methodology The number of possible ties and the total number of possible outcomes was calculated with the LINGO model shown in the Appendix A. The basic approach is a standard one: When faced with an apparently difficult problem, solve an even bigger and more general problem, such that the solution to our specific problem falls out as a special case. The special problem is to count the number of ways in which 51 states(washington, DC counts as a state as far as the electoral college is concerned) can cast their electoral votes so that the Democrats get exactly 538/2 = 269 votes. We assume that only two parties get votes. The general problem that is solved(by dynamic programming) is: For j = 1 to 51: For i = 1 to 269: Compute WAYS( i, j) = the number of ways that the Democrats can get exactly i electoral votes using just the first j states. We are interested in the special case: i = 269, j = 51. If state j has V(j) electoral votes and state j is an all or nothing state, then WAYS( i, j) = WAYS( i, j-1) + WAYS( i-v(j), j-1). In case one is interested in generating an example vote configuration that results in a tie, the Appendix B contains a What sbest spreadsheet model that allows one to easily generate tie solutions, perhaps with additional conditions such as: If California votes Republican, is it still possible to have a tie?. Statistical Approaches When all states can split their electoral votes and each vote has probability.5 of going to the Democrats, then the number of votes for the Democrats has a binomial distribution with n = 538, p =.5. The probability of a tie is then easily computed as above to be The Normal approximation to the binomial is good in this case. The mean is 269, with a standard deviation of This gives the probability of a tie as One can also use the Normal approximation when all states vote all-or-nothing. The mean is still 269, however, the standard deviation is This gives the probability of a tie as Note, this is slightly off from the exact probability of computed above. Appendix A! LINGO model to count the number of ways that a tie can result in U.S. Presidential Election Electoral College. LINGO can be downloaded from ; SETS: STATE: VOTES; KNT/1..269/:; KXS( KNT, STATE): WAYS; ENDSETS 2

3 DATA: STATE, VOTES = Maine 4 Nebraska 5 Alabama 9 Alaska 3 Arizona 8 Arkansas 6 California 54 Colorado 8 Connecticut 8 Delaware 3 DC 3 Florida 25 Georgia 13 Hawaii 4 Idaho 4 Illinois 22 Indiana 12 Iowa 7 Kansas 6 Kentucky 8 Louisiana 9 Maryland 10 Massachusetts 12 Michigan 18 Minnesota 10 Mississippi 7 Missouri 11 Montana 3 Nevada 4 New_Hampshire 4 New_Jersey 15 New_Mexico 5 New_York 33 North_Carolina 14 North_Dakota 3 Ohio 21 Oklahoma 8 Oregon 7 Pennsylvania 23 Rhode_Island 4 South_Carolina 8 South_Dakota 3 Tennessee 11 Texas 32 Utah 5 Vermont 3 Virginia 13 Washington 11 West_Virginia 5 Wisconsin 11 Wyoming 3 ; ENDDATA!Definition: WAYS( I, J) = number of ways of get a Democrat electoral count of I, 3

4 using only the first J states;! Get started with Maine and Nebraska. They are special in that they can split their KNT( I) I #LE# VOTES( 1): WAYS( I, 1) = KNT( I) I #GT# VOTES( 1): WAYS( I, 1) = 0! If Maine is an all or nothing state, replace above by this;! WAYS( VOTES( 1), 1) = KNT( I) I #NE# VOTES( 1): WAYS( I, 1) = 0! Now do KNT( I): WAYS( I, 2) = WAYS( I, 1) KNT( K) I - K #GT# 0 #AND# K #LE# VOTES( 2): WAYS( I-K,1))! Now the rest of the STATE( J) J #GT# KNT( I) I #LT# VOTES( J): WAYS( I, J) = WAYS( I, J - 1 WAYS( VOTES( J), J) = WAYS( VOTES( J), J-1) + 1;! We can get to I,J either without this state or with KNT( I) I #GT# VOTES( J): WAYS( I, J) = WAYS( I, J - 1) + WAYS( I - VOTES( J), J - 1)! Finally, number of ways of getting 269 votes using all states; TIES = WAYS( STATE)! Number of possible vote outcomes of any kind; ALL = (VOTES(1)+ 1)* STATE) -2! If Maine and Nebraska are all or nothing states, turn this on;! ALL = STATE! It may be of interest to look at the numbers in a spreadsheet. Turn on the following to send results to an appropriately named spreadsheet containing appropriately pre-defined "C:\P123\ELECTORAL.XLS") = "C:\P123\ELECTORAL.XLS") = ALL;!ENDDATA; 4

5 Appendix B A What'sBest model for finding tie solutions to the Electoral College vote in the U.S. Presidential Election. Note What'sBest is downloadable from:electoral State Votes Go for Dems? Alabama 9 0 Alaska 3 0 Arizona 8 1 Arkansas 6 0 California 54 1 Colorado 8 1 Connecticut 8 1 Delaware 3 0 D.C. 3 0 Florida 25 0 Georgia 13 1 Hawaii 4 0 Idaho 4 0 Illinois 22 0 Indiana 12 0 Iowa 7 1 Kansas 6 0 Kentucky 8 1 Louisiana 9 0 Maine1 1 0 Note, Maine is special in that Maine2 1 0 it may split its 4 electoral votes. Maine3 2 0 Maryland 10 0 Massachusetts 12 0 Michigan 18 0 Minnesota 10 0 Mississippi 7 1 Missouri 11 0 Montana 3 0 Nebraska1 1 0 Note, Nebraska is special in that Nebraska2 2 0 it may split its 5 electoral votes. Nebraska3 2 0 Nevada 4 0 New Hampshire 4 0 New Jersey 15 1 New Mexico 5 0 New York

6 North Carolina 14 1 North Dakota 3 1 Ohio 21 0 Oklahoma 8 1 Oregon 7 1 Pennsylvania 23 1 Rhode Island 4 0 South Carolina 8 1 South Dakota 3 1 Tennessee 11 1 Texas 32 1 Utah 5 0 Vermont 3 1 Virginia 13 1 Washington 11 0 West Virginia 5 1 Wisconsin 11 1 Wyoming 3 0 Total= 538 For dems 269 Constrain the votes for Dems to equal 538/2----> = To tie 269 6

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