Note to the Teacher: ..........
Prerequisites: Chapter 2, Election I, and Chapter 8.
This is a continuation of Election I, see there for the description of the games considered. Back there, not much could be said for the many versions who don't have pure Nash equilibria, but now the concept of mixed strategies allows us a fresh look on these games. Since we have two-person zero-sum games, von Neumann's and Robinson's theorems apply, and these mixed Nash equilibria are really meaningful. We will use the Excel sheet Election2.xls, where the payoff matrix is calculated automatically, and where Brown's fictitious game method to approximately find mixed Nash equilibria is implemented.
The chapter is structured by a few statements about campaigns, most of them rather obvious, but most of them also supported by some empirical evidence from data collected by the organization FairVote [FairVote] reporting attention---visits or money spent---in the rather close 2004 Bush/Kerry campaigns. We will then analyze concrete simple examples to see whether such statements are, in principle, also supported by Game Theory in our small and simple models.
Before we start with the statements, let us finish the analysis of the second example ELECTION(7,8,13|-1,-1,1|4,4) from the first part, which we couldn't finish then and there since this game doesn't have a Nash equilibrium in pure strategies. Remember that after repeatedly eliminating weakly dominated moves, we arrived at this matrix:
| 0,0,4 | 1,0,3 | 0,1,3 | 1,2,1 | |
| 1,0,3 | -1 | -1 | 1 | -1 |
| 0,2,2 | -1 | 1 | -1 | -1 |
| 0,3,1 | -1 | -1 | -1 | 1 |
| 2,2,0 | 1 | -1 | -1 | -1 |
Since Beth's moves (1,2,1) and (2,2,0) perform identically against Ann's four remaining moves, MixA forms a Nash equilibrium with each one of Beth's mixes of 25% of (0,0,4), 25% of (1,0,3), 25% of (0,1,3), x% of (1,2,1), and (25-x)% of (2,2,0).
The report mentioned above [FairVote] presents data---money spent on ads, and number of visits
from candidates for president and vice president
during the last 37 days before the election, per state.
The data can also be found in this Excel sheet.
Many states didn't receive any attention in the form of visits or money for ads,
but obviously those states were considered to be already lost or won.
Only the "swing states", also called "battleground states", where the standing was close
did attract effort. It would be interesting to find out which states were considered swing states
at that time---Florida, where the result was too close to call for days in the 2000 election was certainly
among them.
But since we don't know the pre-election perception of the parties,
we take a post-election point of view and just look
at those states where the results for Bush or Kerry differed by at most 5%.
For these states the number of visits versus the number of electoral votes are displayed
in the chart to the right, in blue for the democrats and in red for the republicans.
The regression lines, the straight lines that are "closest" to the data points,
are also displayed.
Since these lines (almost) contain the origin,
the relationship between the variables is close to proportional---
twice the number of votes attracts twice the number of effort.
Let us try to confirm this pattern in our small model, in cases where all three districts have different size and the sum of votes in the smaller districts exceeds the number of votes in the largest district, which we called "7-8-13" in Election I. Although a player winning any two districts wins the presidency, if one district has a tie, the candidate winning the larger district wins the presidency.
Using the Excel sheet Election2.xls, and running Brown's fictitious play 5000 times, we find mixed strategies for the symmetric games ELECTION(7,8,13|0,0,0|3,3), ELECTION(7,8,13|0,0,0|4,4), and ELECTION(7,8,13|0,0,0|5,5). In ELECTION(7,8,13|0,0,0|3,3), both Ann and Beth use 20% of each of (0,0,3), (1,0,2), (0,1,2), (1,1,1), and (1,2,0). Therefore, with 20% probability a player puts 0, 1, 0, 1, respectively 1 resources into the smallest district C. Therefore the expected value of resources put into district C by each player is 20%·0 + 20%·1 + 20%·0 + 20%·1 + 20%·1 = 0.6. In the same way, each player puts 20%·0 + 20%·0 + 20%·1 + 20%·1 + 20%·2 = 0.8 resources into district D, and 20%·3 + 20%·2 + 20%·2 + 20%·1 + 20%·0 = 1.6 resources into the largest district E. Therefore both put on average 20% of their three resources into district C, 27% into district D, and 53% into district E.
Of course, these values simplify, maybe oversimplify, matters. In particular, it is not enough to tell the players the percentages how frequently resources should be put into the different districts. One can find mixed strategies with the same percentages than the two mentioned above that still give rather different payoffs.
Doing the same for the other two games, we find similar values in ELECTION(7,8,13|0,0,0|4,4): Both players put on average 23% of their three resources into district C, 27% into district D, and 51% into district E. For ELECTION(7,8,13|0,0,0|5,5) the numbers are closer together, as they are 25%, 30%, and 45%. Having more resources, more time, weakens the emphasis on the large district.
Do we also observe the proportionality between effort and number of votes? Yes, in all three games, but the question may be ill-posed, since the same payoff matrices, the same solutions, and the same distributions of effort into the three district occurs in any case where all three districts have different number of votes, and the two smaller districts have more than the large one. So, we get the same effort distributions as above for 2, 12, 13 votes, or 11, 12, 13 votes, or 7, 8, 13 votes, and the values can not be proportional in all these cases.
ELECTION(7,8,13|0,0,0|4,4): Both Ann and Beth use 9% of (0,0,4), 27% of (1,0,3), 9% of (0,1,3), 9% of (0,2,2), 9% of (2,1,1), 27% of (1,2,1), and 9% of (2,2,0).
Now let's look at asymmetric games where Ann has an advantage in one district and Beth in another. Should a player put more resources into the district where she has an advantage, or should she rather try to turn the district where she is behind? We use the same method as in the previous section, finding first mixed Nash equilibria, and then adding the expected resources in the different districts for these mixed strategies.
Consider ELECTION(7,8,13|0,-1,1|3,3), for instance, where Ann is one resource behind in district D and leads by one resource in district E. One mixed Nash equilibrium found by Brown's fictitious play in the Excel sheet consists of about 9% of (0,0,3), 18% of (1,0,2), 18% of (0,1,2), 18% of (0,2,1), and 37% of(1,2,0) for Ann, and 26% of (0,0,3), 13% of (0,1,2), 28% of (0,1,2), 8% of (1,1,1), 19% of (2,1,0), and 5% of (1,2,0). On average, Ann puts 18%, 42%, and 39% of her resources into districts C, D, and E, and Beth puts 22%, 22%, and 56% of her resources into districts C, D, and E. The expected payoff for Ann in this mixed Nash equilibrium is about 0.091. Thus Ann has an advantage, which is obvious since the district where she leads is larger than the district where Beth is ahead. Moreover Ann puts most effort into district D, where she is behind, and Beth puts most effort into district E, where Beth is behind.
ELECTION(7,8,13|0,-1,1|4,4) confirms this picture: Ann puts 26%, 40%, and 34% of her resources into districts C, D, and E, and Beth puts in 23%, 17%, and 60% of her resources. The expected payoff for Ann equals 0.021 in this case. However, for a higher number of available resources, the patterns seems to shift, as in ELECTION(7,8,13|0,-1,1|5,5) the numbers are 24%, 38%, 38% for Ann, and 23%, 24%, 53% for Beth. Ann seems to shift from attacking at district D to defending at district E. Also, Ann's advantage diminishes further to a value of only 0.008.
Based on these cases, we may dare to suggest that Ann and Beth should on average put more effort into the districts where they have a disadvantage, at least if the number of resources still available is small.
What about the real world? Look at the largest swing states Florida, Pennsylvania, Ohio, Michigan, and look also at Wisconsin and Iowa which got more visits than the remaining states. Bush won in Florida, Ohio, and Iowa, Kerry won in Pennsylvania, Michigan, and Wisconsin. In each of these six states the candidate who invested less effort did win. If we assume that all these states have been considered swing states before, an assumption certainly confirmed for Florida, as explained above, but also supported by the fact that the candidates put so much effort into these states, we can just draw two conclusions: Either people dislike politicians so much that more effort means less votes, an assumption which sometimes has some merits but overall still seems too cynical. Or it means that in these states the candidate investing more was initially behind, but just didn't quite make it. But then, we have a confirmation of the above thesis that candidates put more effort into states where they are a little behind.
As an exercise, use this Excel sheet to check whether looking on the money spent on ads confirms this too.
ELECTION(7,8,13|-1,1,0|3,3) is an interesting game in several respects: First, it has many mixed Nash equilibria. Let us present the payoff matrix:
| 0,0,3 | 1,0,2 | 0,1,2 | 2,0,1 | 1,1,1 | 0,2,1 | 3,0,0 | 2,1,0 | 1,2,0 | 0,3,0 | |
| 0,0,3 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 |
| 1,0,2 | -1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | -1 | 1 |
| 0,1,2 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 |
| 2,0,1 | 1 | -1 | -1 | 1 | 0 | -1 | 1 | 1 | 1 | 1 |
| 1,1,1 | -1 | -1 | -1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 |
| 0,2,1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 3,0,0 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 0 | -1 | -1 |
| 2,1,0 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 0 | -1 |
| 1,2,0 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 0 |
| 0,3,0 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 |
Take now a mix of x of (0,1,2), x of (2,0,1), y of (0,0,3), and y of (1,0,2) for Ann, where 2x+2y=1. Such a mix achieves the following payoffs against the pure moves of Beth:
| 0,0,3 | 1,0,2 | 0,1,2 | 2,0,1 | 1,1,1 | 0,2,1 | 3,0,0 | 2,1,0 | 1,2,0 | 0,3,0 | |
| mix | 0 | 2y | y | 2x+2y | x+2y | 0 | 2x+2y | 2x+2y | 2x-2y | 0 |
As for the amount of resources the players using these mixed strategies invest in the different districts, Ann's x-y mix puts (2x+y)/3 into district C, x/3 into district D, and (3x+5y)/3 into district E. These numbers vary from 25%, 8%, 67% for x=25% to 33%, 17%, 50% for x=50%. On the other hand, Beth's mix puts nothing into district C, 33% into district D, and 67% into district E. Thus the values can vary, but both put most effort into the largest district, and both put more effort into the district where they are behind than into the district where they are leading, as in most of the previous examples.
Is "large and large" better than "middle and small"? Is Ann having a larger, 2-resource lead in the large district better off than Beth who leads in both two smaller districts by one resource each? One might think so, but our models show that this is not the case. The first two games give an expected value of 0.