Electoral College: A Multibattle Contest with Complementarities
February 2017
Preliminary draft
Cary Deck
Sudipta Sarangiƚ
Matt Wiserǂ
Abstract: The Electoral College system used to elect the President of the United States is an
example of a multibattle contest with complementarities. Due of heterogeneity in the value of
the different state level sub-contests, many different combinations of state victories can lead to
an overall victory which impacts contestant strategy. This paper develops a theoretical of such a
multibattle contest with complementarities, and tests its predictions in the lab for a simplified
four state Electoral College map. To focus on purely on the strategic aspects of the Electoral
College game, in the lab we use three different Electoral College maps while always keeping the
total number of delegates the same. Our experimental results show that subjects behave quite
differently from the theoretical predictions. Instead they seem to be using a mental accounting
rule according to which they choose to spend half the value of the prize as their budget in the
lottery success function. Then, depending on the map, they follow different bidding patterns the
most predominant pattern being one that implicitly takes into account the marginal importance of
each state for winning the election.
Keywords: Multibattle Contests, Complementarities, Electoral College, Experiments
JEL Codes: C7, C9, D7

University of Arkansas and Chapman University. E-mail: [email protected]
Virginia Tech. E-mail: [email protected]
ǂ
University of South Alabama. E-mail: [email protected]
ƚ
1. Introduction
The Electoral College system used to determine the President of the United States has received a
lot of attention lately, with questions ranging from its usefulness, to campaign strategies of the
candidates. Our paper treats the Electoral College as a classic example of a multibattle contest and
focuses on strategic behavior in this contest using theory and experiments.1 In this contest,
presidential candidates compete in each state (and the District of Columbia but not in US territories
such as Puerto Rico and Guam) for delegates. The number of delegates in a state is based on its
number of members in Congress and ranges from 3 in less populated states like Delaware and
North Dakota up to 55 in California. In total there are 538 delegates and whoever claims a majority
of delegates wins the election. The key question here is that since that many different combinations
of states could provide a majority of the delegates, how should the plan their campaign to win the
elections.
In the paper we set up a simple model of a multibattle contest consisting of n different states each
with a specific number of delegates. The two players in the contest simultaneously decide how
much they wish to spend or bid on each state. The lottery success function is used to determine the
winner of each state, and both players pay their bids regardless of who wins the state. The winner
of the contest is the player who wins a majority of the delegates. We then test this model in the lab
using a setup that allows for four states with different possible combinations of delegates, while
keeping the overall number of delegates unchanged. Since the optimal behavior for different
combinations of delegates is different, this allows us to study the strategic behavior of the players
in the lab.
There is a substantial literature on US Presidential campaign starting with the seminal work of
Brams and Davis (1973). A few years later, Lake (1979) argued that it more reasonable to assume
that the candidates wish to maximize the probability of winning the election instead of trying to
win the expected popular vote. Although not quite described in such terms, today it would be best
to describe Lake’s work as a stochastic asymmetric “majority rule” Colonel Blotto game. The
involvement of multiple states in the contest and the majority rule requirement clearly makes the
Electoral College a majority rule Colonel Blotto game. The stochastic part refers to the fact that
the Tullock style lottery success function is used to determine the outcome of the game. It is
asymmetric since different states or battles are different values or delegates in the contest. Lake
(1973) assumed that the two players had identical budgets. In a recent paper, Duffy and Matros
(2015), generalize Lake’s paper by allowing for non-identical (but sufficiently similar) budgets,
and provide results for up to four states. They show through an example that it is not always
possible to find a solution to the Electoral College competition when there are five or more states.
Their key finding is that the bids for each state are proportional to its Banzhaf (1965) power index.
By contrast, we are able to solve for any number of states and show that the bids are the optimal
1
A multibattle contest is one where the overall winner is determined by the outcome of a set of sub-contests.
1
Tullock bids weighted by the Banzhaf index. This is possible in our model since we do not impose
an external budget constraint, an assumption that may be considered realistic since modern election
campaigns continually keep raising funds.
We then test this in the lab by considering a map of four states with a total of seven delegates. We
use three different maps of the four states that all add the same (2+2+2+1 = 4+1+1+1 = 3+2+1+2
=7) to test our model. Note that the theoretical prediction for each map is quite different. Not
surprisingly, this is a difficult game for the subjects. However, we find that subjects seem to have
a systematic behavioral pattern across all three maps. Besides being able to test the theoretical
predictions of our model, there is another reason for taking the Electoral College game to the Lab.
The empirical literature on campaign resource allocation provides a number of reasons for why
candidates might choose to allocate resources in a particular state. Apart from victory
considerations, these might include the desire to help another candidate, or possibly because
devoting resources to a particular state might raise more campaign funds or to influence the
political socialization process (see for instance Shaw, (1999) and Gimple, Kaufmann and PearsonMerkowitz, (2007)). Most of these empirical studies have to resort to econometric techniques to
untangle to strategic effects from the non-strategic effects. By allowing us to control the
environment, the lab is an excellent complement to studying how strategic considerations drive
behavior without being confounded by other factors.2 To the best of our knowledge this is the first
paper to study this phenomenon in the lab.
It is also important to explain how this paper relates to the rest of the multibattle contests literature
(see Konrad 2009; and Kovenock and Roberson 2010). While the literature on multi-battle contests
goes back to Borel (1921) and Borel and Ville (1938) (see also Gross, 1950; Gross and Wagner,
1950; and Freidman, 1958), most of the previous work on multi-battle contest has used a simple
majority of the sub-contests (or a symmetric values) rule. In a presidential US election this would
be akin to each state having one delegate. More recently, there has been more interest in other
rules. For example, Clark and Konrad (2007) and Golman and Page (2009) consider a weak-link
structure where one contestant needs to win a single battle to be victorious while the other needs
to win all of the battles. Szentes and Rosenthal (2003a, b) consider a more general case where one
party needs to win some specified fraction of the battles in order to be the overall winner.
However, in most of the existing literature (and unlike the Electoral College), the individual battles
are interchangeable.3 One exception to this norm is Kovenock, et al. (2012) who consider a contest
for a set of geographic regions where there are complementarities between some of the adjoining
regions. Specifically, they exploit a 2  2 version of the game of Hex in which two players attempt
to win combinations of battles in order to form a path across the hexagonal board or map. The key
In fact Shaw (pg. 893, 1999) states that, “The bottomline, then, is that analyses of campaigns are hampered by a
shortage of knowledge with respect to the candidates' goals and allocative behavior.”
3
If the battles occur in sequential order, then the strategic incentives for the battles may differ, but it is still the case
that the identity of the battle does not matter, only its order in the sequence.
2
2
feature of this a game is that each player has a different set of winning combinations or paths (and
hence complementarities), though some winning combination may work for both players.
Moreover, winning a particular state or hexagonal cell on the board does not guarantee victory in
the game unless it enables the player to form a path across the board. Like our Electoral College
game, Kovenock, et al. (2012) assume that each region is allocated according to a Tullock contest
success function and show that players should place positive bids for every region in equilibrium
as each region is part of some winning combination. Further, regions that have more strategic
value, as they are in more winning combinations, should receive higher bids.
Like the Hex game, the Electoral College game is also a game with complementarities. Winning a
particular state is important only when viewed as a part of a combination with the other states that
allows the player to win a majority of the delegates. However, the Electoral College game differs
from the Hex game in one crucial way – in this game obtaining is majority requires winning the
same set states for both players, while in the Hex game each player has their own winning path in
the hexagonal 2  2 board. By varying the distribution of the delegates across states, the strategic
relevance of the different states and hence their equilibrium bids change. In particular, two
different regions can have the same strategic value even though they have different numbers of
delegates. Further, the strategic value of a region can change if delegates are reallocated among
other regions. Finally, the total expenditure is shown to vary with the allocation of delegates
despite the total number of delegates and the prize remaining fixed.
In a recent paper, Deck, et al (2016) design an experiment to test the predictions of Kovenock, et
al. (2012). While the behavioral results are largely consistent with the theoretical model in
aggregate, they conclude that the model does not capture strategic behavior in the game. Instead,
subjects tend to compete on minimal winning sets of regions, combinations of regions that are
sufficient for victory but such that no proper subset would be. This paper also presents the results
of laboratory experiments that test the theoretical predictions. We find that the Electoral College
game like the game of Hex is a difficult game. As a prelude to the results, players do not seem to
be behaving as predicted by theory. The most popular strategy seems to be bidding everywhere
and which is in sharp contrast to the Hex game where bidding on minimal strategies is the most
common strategy.
The rest of the paper is organized as follows. Section 2 sets up the theoretical model and provides
predictions for the experiments. Section 3 has the experimental set up and the results are presented
in Section 4. Section 5 concludes.
2. Model and Hypotheses
Our model set up and notation for the Electoral College contest closely follows Duffy and Matros
(2015). There is one key difference however: we do not have an explicit budget constraint, though
3
our players do incur a cost for each region they bid on. This difference allows us to improve on
the results of Duffy and Matros (2015).
Let there be two players denoted by x and y. We have a set of N = {1, 2, …, n} items (or states).
Item i is commonly valued at Wi > 0 by both players. The total value of all n items is given by
𝑊 = ∑𝑖∈𝑁 𝑊𝑖 . In the context of the Electoral College, Wi can be interpreted as the number of
delegates associated with state i. Each player wishes to obtain states to have a majority of the
delegates. Player x’s strategy is an n-dimensional vector (x1, x2, …, xn) that specifies how much
the player wishes to spend or place as a bid on state n.4 Player y has a similar strategy. The lottery
success function is used to denote which player obtains the state i. The probability that the state
𝑥𝑖
goes to player x is given by 𝑥 +𝑦
. Thus each state can only be won by one player. The lotteries are
𝑖
𝑖
assumed to be statistically independent.5 Both players place their bid on the n states simultaneously
and nature determines the winner of each region. Note that regardless of who wins a region, both
players have to pay their bid values.
Each possible subset of states from the set N can be represented by an n-dimensional characteristic
vector t = (t1, t2,…, tn) where ti {0, 1} for any i = 1, 2, …, n. When ti = 0, the state does not belong
to the set and vice versa. The rest of the paper uses the corresponding characteristic vector to
denote subsets. Let V be the set of winning subsets under the objective function where the winner
is the player who wins more than half of the number of delegates. Then a subset t  V, is a winning
𝑊
subset if ∑𝑛𝑗=1 𝑡𝑗 𝑊𝑗 > 2 . Next denote by Vi a set of winning subsets where state i is pivotal and by
Piv(i) the number of winning subsets in which state i is pivotal, i.e., Piv(i) = || Vi||.
A player is the winner of the electoral competition game if she obtains a winning set. We assume
that the prize from winning the contest is worth V and the losing the contest earns a payoff of 0.
Player x maximizes her chance of obtaining a winning subset by solving the following
maximization problem:
𝑛
𝑥𝑗
𝑦𝑘
𝑚𝑎𝑥 ∑ ∏
∏
− ∑ 𝑥𝑖
𝑥1 ,…,𝑥𝑛
𝑥𝑗 + 𝑦𝑗
𝑥𝑘 + 𝑦𝑘
𝑡∈𝑉 𝑗:𝑡𝑗 =1
𝑘:𝑡𝑘 =0
(1)
𝑖=1
𝑥
𝑗
where ∏𝑗:𝑡𝑗=1 𝑥 +𝑦
is the probability of winning all the items that belong to the subset t, and the
𝑗
𝑗
second term is the probability of losing all the items that do not belong to t. The last term is the
total bid of player x. Player y solves the opposite maximization problem given by
𝑛
𝑦𝑗
𝑥𝑘
𝑚𝑎𝑥 ∑ ∏
∏
− ∑ 𝑦𝑖
𝑦1 ,…,𝑦𝑛
𝑥𝑗 + 𝑦𝑗
𝑥𝑘 + 𝑦𝑘
𝑡𝑉 𝑗:𝑡𝑗 =1
4
5
𝑘:𝑡𝑘 =0
(2)
𝑖=1
We use the words item, state and region interchangeably.
If both players bid 0 for item i, then each has an equal probability of winning the item.
4
where each term has an explanation similar to that given for equation (1).
Following Duffy and Matros (2015), we make an assumption that guarantees a unique winner in
the Electoral College contest for all realizations of the individual lotteries.
Assumption 1: ∑𝑛𝑗=1 𝑡𝑗 𝑊𝑗 ≠
𝑊
2
for any subset t.
We also need the following definition.
Definition 1: An item i is pivotal in subset (t1, …, ti-1, 1, ti+1, …, tn) if (ti=1, t-i) is a winning subset
but (ti=0, t-i) is a losing subset.
A state is pivotal if removing it from the set, while keeping the membership of all the other states
in the set unchanged, changes the winning subset to a losing subset. We can now state our main
result.
Proposition 1: In an Electoral College contest over n states for a prize worth V, the Nash
𝑉 𝑃𝑖𝑣(𝑖)
.
2𝑛−1
equilibrium bid of each player for state i is given by 4
Proof: From equations (1) and (2) it follows that we have a symmetric game. So, 𝑥𝑖 = 𝑦𝑖 making
it equally likely for either player to win state i. Thus, for each player, they will have an identical
1
probability of winning each subset from the other n-1 states. From Definition 1, it follows
2𝑛−1
that,
𝑃𝑖𝑣(𝑖)
2𝑛−1
is the probability that winning state i will be the difference between winning and losing
the overall contest. Hence, the expected value of winning i is 𝑉
𝑃𝑖𝑣(𝑖)
,
2𝑛−1
and thus Player x will be
attempting to maximize the function
(𝑉
𝑃𝑖𝑣(𝑖)
𝑥𝑖
− 𝑥𝑖
)
𝑛−1
2
𝑥𝑖 + 𝑦𝑖
with Player Y maximizing a similar function. From the first order condition, we obtain
𝑉
𝑃𝑖𝑣(𝑖)
𝑦𝑖
2𝑛−1 (𝑥𝑖 +𝑦𝑖 )2
= 1.
These are clearly continuous and concave functions and since, 𝑥𝑖 = 𝑦𝑖 , as in the standard Tullock
𝑉 𝑃𝑖𝑣(𝑖)
2𝑛−1
contest, this gives us an optimal bid of 4
on each state i in the Nash equilibrium.
□
Note that the equilibrium bid for each state can also be seen as a weighted version of the
equilibrium bid in Tullock contest. In the standard Tullock contest, the optimal bid for a prize
worth V is given by V/4. In our optimal bid, this value is weighted by the probability of winning
that state being the difference between victory and defeat, given by the second term
𝑃𝑖𝑣(𝑖)
.
2𝑛−1
Next,
observe that unlike Duffy and Matros (2015), we are able to solve the Electoral College contest
5
for an arbitrary number of n states, rather than being limited to n < 5. Of course, as already pointed
out we are able to do this because we do not have an explicit budget constraint which significantly
changes the calculation of pivotal voters. Example 1 in Duffy and Matros (2015) shows that for n
= 5 states, a solution does not exist in their model. Below, we solve the same example using our
model.
Example 1 (Example 1 Duffy and Matros (2015), pg. 7): Let n = 5 and 𝑊1 = 3, 𝑊2 = 𝑊3 = 𝑊4 =
𝑊5 = 1. Since we have a total of 7 electoral votes, there can be no ties. It is easy to check that there
are 16 winning subsets, in which 𝑊1 = 3 will be pivotal in all but 2 subsets (the sets consisting of
all five states, and of the four states each worth one vote). Each 𝑊𝑖 = 1 is pivotal twice (in the
subset consisting of the four one vote states, and the set consisting of a single one vote state
combined with the three vote state). Let the value of the prize be V. In equilibrium we will see
𝑉 14
7𝑉
𝑉
2
𝑉
bids of (4 ) 16 = 32 on the state worth three votes, and (4 ) 16 = 32 on the other four states.
□
Next we solve an example that forms the basis of our experimental design.
Example 2: Let n = 4, again with a total of 7 electoral votes. Suppose all states are restricted to
having a positive number of delegates, then (up to
relabeling) there are only three possible combination of delegates: (4,1,1,1), (3,1,1,2), and
(2,2,2,1). For convenience, and to help explain our experimental design, the vector m = {𝑑𝑁𝑜𝑟𝑡ℎ ,
𝑑𝑊𝑒𝑠𝑡 , 𝑑𝐸𝑎𝑠𝑡 , 𝑑𝑆𝑜𝑢𝑡ℎ } will be referred to as an Electoral College map or simply a map. Figure 1
shows the maps with the delegates mentioned above.
Figure 1. Electoral College Maps
dNorth = 4
dWest = 1
dEast = 1
dSouth = 1
(4,1,1,1)
dNorth = 2
dNorth = 3
dWest = 1
dEast = 1
dSouth = 2
(3,1,1,2)
dWest = 2
dEast = 2
dSouth = 1
(2,2,2,1)
We will now consider each map separately. We will also assume the prize is worth V to each
player. Notice that in stark contrast to Kovenock et al. (2015) and Deck et al. (2016) the set of
winning sets is the same for both players, and in any partition of states exactly one player will be
the winner.
Case 1: In the case of (4,1,1,1), the solution is simple. The state with 4 delegates is pivotal to
every winning subset, and thus the optimal bids will be V/4 for this state. Since the remaining
three states are not pivotal, the optimal bid on them is zero.
6
Case 2: In the (3,1,1,2) case, the state with 3 delegates is pivotal in 6 of the 8 winning subsets, not
being pivotal in the subset consisting of all the elements and not being part of the (1,1,2) winning
𝑉6
3𝑉
subset. Thus the optimal bid on this state is 4 8 = 16. Each of the other states is pivotal in 2 winning
subsets, those consisting of the state with the 3 delegates or the (1,1,2) subset, and thus the optimal
bid is 3 on these other states, for a total bid of 3V/8.
Case 3: Finally, in the (2,2,2,1) case, the state with 1 delegate is never pivotal, while the states
with 2 delegates are pivotal in 4 subsets each, consisting of being paired with exactly one of the
other 2 states, with and without the 1 delegate state. Thus the optimal bids are V/8 on each of the
two delegate states, again for a total bid of 3V/8.
The information from the three cases is summarized in Table 1 below. We also define the notion
of a minimal wining set in this table, which helps the reader compute the minimum (but not
optimal) investment necessary to win a given map. A minimal wining set is an element of V such
that no proper subset of it is also in V. The minimal winning sets for each map shown in Figure 1
are presented in Table 1.
Table 1. Theoretical Predictions by Map
Map
Minimal Winning Sets
(4,1,1,1)
{North}
(3,1,1,2)
{North, West},
{North, East},
{North, South},
{West, East,
South}
(2,2,2,1)
{North, West},
{North, East},
{West, East},
Equilibrium Investment
North
West
East
South
Total Investment
Expected Profit
V/4
0
0
0
V/4
V/4
3V/16
V/16
V/16
V/16
3V/8
V/8
V/8
V/8
V/8
0
3V/8
V/8
The equilibrium predictions for these three maps reveal several noteworthy results which provide
interesting points to study in the experiments: (1) The strategic value of a state is not solely a
function of its number of delegates. (2) On a map the strategic value of a state is not a strictly
monotonic function of its number of delegates. (3) Despite the fact that the maps all have the same
total number of delegates, the total investment is not constant. Since each player places the same
bid, in equilibrium each expects to win half the time and generate revenue of V/2 and as a result
the expected profit can vary (see Table 1).
7
To discuss how our results can be related to lake (1979) and Duffy and Matros (2015) we need to
define the notion of Banzhaf Power Index (Banzahf, 1965).
Definition
2:
𝐵𝑃𝐼(𝑖) =
The
Banzhaf
𝑃𝑖𝑣(𝑖)
𝑃𝑖𝑣(1)+⋯+𝑃𝑖𝑣(𝑛)
=∑
Power
∑𝑡𝜖𝑉𝑖 1
Index
(BPI)
can
be
defined
as
follows:
.
𝑡𝜖𝑉1 1+⋯+∑𝑡𝜖𝑉𝑛 1
Remark 1: Electoral Competition with and without budgets. A key distinction between our paper
and Duffy and Matros (2015), or for that matter Lake (1979) is the presence of an explicit budget
in their papers making a Colonel Blotto game. Additionally, unlike Lake (1979), Duffy and Matros
(2015) assume unequal budgets for the two players, limiting their ability to solve for a game that
has more than 4 states. In our game on the other hand, players do not have a pre-set budget, but do
have to pay their bids on each state. Obviously, this bid sum can be used to define a budget to
facilitate comparison with other two papers. Also, given that both players have identical
preferences over the states in our game, they will both bid the same amount on each state in
equilibrium. Hence the obvious comparison will be with Lake (1979) where players identical
budgets. We now examine conditions under which the players will bid the same amount on each
state in the two models.
Consider Example 1 again, where n = 5 and 𝑊1 = 3, 𝑊2 = 𝑊3 = 𝑊4 = 𝑊5 = 1. Let the prize be
7𝑉
worth V. Then the optimal bid of either player in our model on state 1 is 32 , and on the remaining
7
four states it is 32. In Lake (1979) the optimal bid on State i is given by BPI(i)X, where X is the
budget of either player. Recall that Piv(1) =14 and Piv(2) = Piv(3) = Piv(4) = Piv(5) = 2. So BPI(1)
14
= 22 and BPI(i) =
2
, for i = 2,3,4,5. Hence we see that the bids in our model and Lake’s (1979)
22
model will be identical when the following relationship between the prize value and the budget
holds: 𝑉 =
32𝑋
11
.
Next, we want to compare results with Duffy and Matros (2015). Given that in this game solutions
do not exist when n > 5, let us consider the case shown in the second map of Figure 1, where n =
4, and 𝑊1 = 3, 𝑊2 = 𝑊3 = 1, 𝑊4 = 2. Assume that players have unequal budgets, i.e., X  Y, and
let X < Y without loss of generality. Also following Duffy and Matros (2015), let 𝐾𝑖 =
∑𝑡∈𝑉𝑖 (∏𝑗≠𝑖;𝑡𝑗 =1 𝑋 ∏𝑘≠𝑖;𝑡𝑘=0 𝑌). This expression in the brackets says that the power on X is the
number of other elements in each subset where i is pivotal, and the power on Y is the number of
elements which are not in the subset. Then 𝐾𝑖 is the sum of these for each subset in which state i
is pivotal. Duffy and Matros (2015) use this to generalize the Banzahf Power Index by allowing
the index for state i is obtained by 𝐾𝑖 by the sum of all the 𝐾𝑖 as in Definition 2. Then as in Lake
(1979), this generalized index is multiplied by each player’s budget to obtain their equilibrium bid
on a state. Substituting into the definition we find that 𝐾1 = 3𝑋 2 𝑌 + 3𝑋𝑌 2 , 𝐾2 = 𝐾3 = 𝐾4 =
𝑋 2 𝑌 + 𝑋𝑌 2 . From this it follows that player X will bid X/2 and player Y will bid Y/2 on state 1,
and they will each bid a sixth of their respective budgets on the remaining states provided
𝑌
<
𝑋
11
9
,
8
with this last constraint being a requirement to satisfy second order conditions. Obviously, direct
comparison between the two models is not possible since bids are identical in our model. But X <
Y, we can see that 𝑉 =
8𝑌
3
for the high budget player to make the same bids as in our model.
3. Experimental Design
To test the predictive success of the equilibrium analysis presented in section 2, a within subject
experimental design was used. In groups of ten, subjects entered the lab, read a set of instructions
(available in the appendix along with a summary statement that was read aloud), and then
participated in a series of election contests. Subjects completed three blocks of 10 contests, using
a different map in each block. Subjects were randomly and anonymously matched at the start of
each contest. After a contest was completed, the subjects observed the outcome for each state and
their own profit, but not the investment or profit of their counterpart. Data were collected from
six groups of subjects and each group experienced the three maps in a unique sequence to control
for order effects. Subjects were not informed of how many contest they would complete with a
given map or the number of maps they would face.
In each contest, the prize for claiming the most delegates was set at V = 48. The numeric
predictions for each map are given in Table 2. However, previous experiments have consistently
found that subjects bid approximately twice the equilibrium value or alternatively half the prize
amount in a standard Tullock auction (see Dechenaux, et al. 2012). Further, the results of Deck,
et al. (2016) suggest that subjects facing a contest with complementarities behave as if they are
using a two stage rule of first fixing the bid amount which equals half the value of the prize, and
then deciding how to allocate it across the different contests. Moreover, a majority of the times
subjects were found to be placing bids on minimal winning sets. Taking this behavioral pattern
into account leads to the behavioral predictions shown in Table 2. For example, in (3,1,1,2) North
is in three minimal winning sets each containing two states. If subjects follow the same behavioral
pattern as in Deck, et al. (2016) then the average investment in North would be ¾(12) + ¼(0) = 9,
while the average investment in South would be ¼(12) + ¼(8) + ½(0) = 5. While the standard
theoretical model and the behavioral hypotheses generally lead to different predictions, both
approaches predict the same investment in North for the (3,1,1,2) map and in cases where the state
is not part of a minimal winning set.
9
Table 2. Theoretical and Behavioral Predictions
Map
(4,1,1,1)
Theoretical Behavioral
North
12
24
West
0
0
East
0
0
South
0
0
(3,1,1,2)
Theoretical Behavioral
9
9
3
5
3
5
3
5
(2,2,2,1)
Theoretical Behavioral
6
8
6
8
6
8
0
0
Each subject received a participation payment of $5 for the one hour experiment, as is standard in
the Economics Laboratory at the University of Alaska-Anchorage, where the experiment was
conducted. In the experiment, all monetary amounts were denoted in Lab Dollars. Subjects began
a session with a balance of 150 Lab Dollars to which their profits and losses in each election were
added. At the end of the experiment, Lab Dollars were converted into US dollars at the rate 25
Lab Dollars = 1 $US. The average payment in the experiment was $11.22.
4. Experimental Results
The results are presented in two subsections. The first examines aggregate behavior while the
second looks at individual bidding strategies.
Aggregate Behavior
Aggregate behavior is summarized in Table 3.6 Several interesting patterns emerge from this table.
The first is that in every case the average observed bid exceeds the theoretical prediction. In the
four cases where a state is never part of a minimal winning set (the predicted bid is 0), several
subjects do place positive bids (on average), but these bids are relatively small and it is important
to keep in mind any bidding error is necessarily one-sided. 7 Note that for the eight cases where a
state is part of a minimal winning set, bidding errors are two-sided. Thus, if people are bidding
according to the theoretical model, the probability that all eight such cases have observed bids
1
above the theoretical prediction is only 28 = 0.004. By contrast, the observed bids are only above
the behavioral predictions in five of the eight cases. The probability of observing an outcome this
or more extreme is 0.727 if people are following the behavioral predictions and making random
6
Recall that subjects experienced each map 10 times. The results are qualitatively unchanged if attention is restricted
to the last half of the observations.
7
Because bids were restricted to be non-negative, when the prediction is 0 errors can only be positive, thus any random
error will lead to observed bids being greater than the prediction.
10
errors. Thus, the aggregate behavior is generally consistent with the behavioral model and not the
theoretical model.
Table 3. Mean and Standard Error of Observed Behavior
Map
(4,1,1,1)
(3,1,1,2)
(2,2,2,1)
21.47
11.60
8.41
North
(1.67)
(0.95)
(0.61)
1.62
4.11
6.60
West
(0.31)
(0.41)
(0.50)
1.78
4.54
7.94
East
(0.37)
(0.42)
(0.58)
1.57
6.22
2.60
South
(0.34)
(0.56)
(0.53)
26.44
26.43
25.54
Total
(1.55)
(1.58)
(1.52)
Averages are based on n = 600 bids. Standard errors are clustered at the subject level.
In the (4,1,1,1) map, West, East, and South do not have significantly different means, with a
Hotelling T-squared of 1.49. In the (3,1,1,2) case we similarly reject the means for East and West
being significantly different with a T-squared of 2.48. However, in the (2,2,2,1) case, though
North, West, and East in theory should be identical, they are actually statistically different, with a
T-squared of 35.44. Table 3 also reveals that for six of the seven cases in which the theoretical and
behavioral predictions differ (see Table 2), the observed behavior is closer to the behavioral
prediction than the theoretical prediction.
A final pattern that emerges from Table 3 is that total spending is nearly identical in the three maps.
This is also supported by regression analysis where an indicator variable is included for each map
and standard errors are clustered at the subject level. An F-test fails to reject that the coefficients
for three maps differ (p-value = 0.701). The similar spending levels and the superior performance
of the behavioral predictions are suggestive of subjects not behaving according to the theoretical
model, rather they seem to be following a heuristic that turns this into a two-stage problem. First,
they seem to be setting the budget of about 24which is identical to the type of mental accounting
suggested in Deck et al. (2016). Second they seem to be deciding how to allocate it to obtain a
majority of the delegates.
The following three graphs (Figures 2, 3, and 4) show the absolute difference between the bids
each player made in each round, and their mean total bid for that particular map (on X-axis). The
frequency of these bids is on the Y-axis. Thus, values near zero show that the players are making
very similar bids between rounds, while high values show larger swings in bidding. We observe
that in each map most subjects’ total amount bid is not very different from the average bid by that
11
subject for that map. In fact in all three cases, the vast majority of the bids are tightly clustered
around 0, suggesting that most players are consistent in their bidding behavior. Note that this
behavior is most pronounced in the (3,2,1,1) map. Keeping this in mind, we now move to the
analysis of individual behavior.
Figure 2: Average bid distribution on (4,1,1,1) Map
12
Figure 3: Average bid distribution on (3,1,1,2) Map
Figure 4: Average bid distribution on (2,2,2,1) Map
13
Individual Behavior
To get a sense of individual behavior we begin by representing subject behavior in a visual manner
by focusing on the distribution of bids, shown here in Figures 5-7. In these graphs, (the size of)
each triangle corresponds to a range of percentages bid on the South, with the points marked by
red dots in each triangle being the breakdown of the remaining percentages between states denoted
by North, East, and West.
We can see distinct patterns in each of the three maps. In the (4,1,1,1) case we see two major
clusters. One of these clusters occurs around the point where 100% of the bids are placed on the
North, which is the equilibrium state, with a secondary cluster near 55% on the North and 15%
on each of East, West, and South, which is indicative of the fact that the proportion of delegates
in each state might be important. In the (3,1,1,2) case we see much less clearly defined clusters,
possibly because every state is important in this map. There are a few areas of note however. In
the 0-10% triangle, we see two clusters along the edges, corresponding to minimal sets consisting
of either North and East or North and West. There is also a large cluster around the point of placing
an even 25% on each of the four states. Finally, we have the (2,2,2,1) map, where we that a large
number of subject bids seem to be occurring the triangle labeled 0-10% on the South. Thus we
see that each of the three maps are played significantly differently by players. Though some
players may be using the same strategies between maps, the differences in the shape and number
of clusters shows there are many players who take different approaches between the different
structures.
14
Figure 5: Distribution of Player Bids (in Percentage Terms) in (4,1,1,1) Map
15
Figure 6: Distribution of Player Bids (in Percentage Terms) in (3,1,1,2) Map
16
Figure 7: Distribution of Player Bids (in Percentage Terms) in (2,2,2,1) Map
17
Although Figures 5-7 provide a visual perspective on how individual bids are placed, we will now
look for explicit patterns in the individual bids in two different ways. First, we will study the spatial
pattern of bidding in the three different maps ignoring the magnitude of the bids placed on each
state. Then we study the bid magnitudes for each state in the three maps.
Spatial placement of bids. We begin by classifying strategies using the pattern of positive bids
placed in each map as a means to check for the intended behavior of subjects. We quantify subjects’
bidding distribution as shown in Table 4. In this table “Everywhere” implies that subjects placed
a bid on all 4 regions, while “Nash” implies that subjects placed bids only on regions supported
by the Nash equilibrium. “Minimal” refers to bids placed only on allowable minimal winning sets,
while “Zero” implies that the subjects did not place a bid at all. All other bidding patterns are
classified as “Other.”
Since magnitudes are not taken into account, this classification actually does not distinguish
between the behavioral and Nash strategy, but it certainly allows us to investigate whether subjects
are using minimal winning sets. The only exception is the (4,1,1,1) case, where both the Nash and
the minimal winning set strategy consists of placing positive bids only on the region with 4
delegates. Table 1 clearly shows that for the (3,1,1,2) and the (2,2,2,1) these two strategies are
different.
Table 4. Classifying bid types by their spatial pattern (in percentage terms)
Type
4,1,1,1
3,1,1,2
2,2,2,1
Everywhere
38.3
60.8
50.3
Nash
45.8
60.8
23.7
Minimal
45.8
21.7
13.8
Zero
4.7
6.0
6.0
Other
11.2
11.5
6.2
Even in the absence of bidding magnitudes this table is quite interesting and it clearly shows that
this a difficult problem for the subjects. Note that a significant number of subjects bid everywhere,
even though Nash and Everywhere coincide only for (3,1,1,2) map. Not surprisingly the highest
number of Everywhere bids are for this case. This is followed by the (2,2,2,1) map where about
half the subjects do not realize that they should bid on the state with 1 delegate. We also find that
bidding on minimal sets is highest for the (4,1,1,1) map where it coincides with the Nash
18
equilibrium. It is lowest in the (2,2,2,1) where there is possibly a co-ordination issue along with
the decision to choose a minimal set.
There is one interesting difference when we compare our outcomes with the Hex game. 8 We find
that bidding on minimal sets is considerably lower in the Electoral College game than in the Hex
game of Deck et al. (2016). This is possibly because in the Hex game, it is easier to focus on one
winning path as the winning paths for the agents are not always identical. In the Electoral College
game, since minimal winning sets are always identical for the subjects, they may choose to focus
on this approach less often. Finally, it is worth emphasizing again that these maps present different
levels of difficulty for the subjects, and as predicted by theory, behavior is indeed different, though
the overall spending on each map is roughly half the value of the prize.
Switching behavior. Given these breakdowns, another natural question is whether individual
subjects have a strong preference for particular type of strategy. In other words, the question is do
we have different types of players? We find that within a map, subjects tend to keep playing the
same strategy 86% of the time after winning money in the previous round. Moreover, if they lose
in a particular round, only 28.3% of the subjects change their strategy with some of them switching
to a different strategy but of the same type. However, there is little evidence of players choosing
the same strategy between maps. Allowing for the possibility of one or two mistakes, there are
only 10 subjects who play “Everywhere” on all three maps, and only 2 who consistently choose
the Nash strategies. Based on these findings, we do not think that there is an inherent player type,
rather subjects are choosing a strategy for each map; these players do vary this choice somewhat,
doing so more frequently after losses.
As expected, players are much more likely to switch strategies after losses. Regardless of bid
magnitudes, there are two distinct ways of analyzing switching behavior of players between
periods of the same map. One option is to simply look at how frequently players switch after losses
and wins. We find that people switch 28.3% of the time after a loss, compared to 12.6% of the
time after a win (they switch 17.7% of the time in the cases where they had zero profit in the
previous round). The switching rates are very similar for the three different electoral maps,
suggesting that this is a fairly robust result. However, this is a somewhat higher switching rate than
in the Hex game of Deck et al. (2016), where switching happened 23.1% of the time after a loss
and 9.4% of the time. Moreover, the analysis of switching behavior suggests that in both games
subjects use a trial and error learning type heuristic. We believe this happens in both games because
after fixing their budgets, subjects have to make an allocation decision for the different battlefields.
The second approach to analyzing switching behavior is to regress a dummy variable for switching
on the variables “period” and “previous profit”. Again, similar to the Hex paper, previous profit is
8
In this section when we refer to the Hex game we mean a contest with complementarities where players do not have
the same objective function over the different battlefields, whereas the Electoral College game implies a contest with
complementarities where players have the same objective function over the different battlefields.
19
negative and significant at the 1% level, but in contrast to the Hex paper period is now also
negative and significant at the 1% level. The values are -0.00158 for previous profit, -0.0118 for
period, and 0.2693 as the constant. Running the same regression in the previous game, for the
simultaneous move games, gives a term of -0.00233 for previous profit, significant at the 1% level,
-0.00164 for period, which is insignificant, and 0.1876 as a constant. Splitting by electoral maps
gives similar results, however drops the significance to the 5% level, possibly because of fewer
observations.
The combination of these results suggest that the players are less certain of their decisions in the
Electoral College game, especially early on because this is when we see a great deal of switching.
Clearly, even though this is a perfectly symmetric game and generally has an easier solution for
optimal play, subjects are having a harder time determining a strategy here. This is primarily
because in the Hex game, winning combinations of subjects could be distinct making it easier to
play them, while here the winning combinations coincide leading to a lot of switching behavior
when subjects are new to the game. In other words in our game players may be more prone to
second guessing themselves after losses, leading to a larger number of switches.
Bid magnitudes. We now move on to analyzing the magnitude of the bidding strategies in each
map. In order to do this we first define three reference points. The first of these will be called Nash
(N) bidding and the value for each of the three maps can be found in Table 2 under the column
labeled “Theoretical”. We define two additional reference points which use the implicit budget of
24 (=V/2) observed in the section on aggregate behavior. One of these is called Behavioral (B)
which implies bidding according the method described in Table 2 under the “Behavioral” column.
Recall that this strategy implies bidding on a minimal number of states needed to obtain majority
and allocates the budget based on the proportion of times a particular state in minimal sets.9 So
the numbers listed under the behavioral column list the average bid on each state, and do not
require subjects to bid on all of them. Finally, the second reference point based on the mental
accounting budget of 24 is called Proportional (P) bidding. This is the analogue of the
“Everywhere” strategy in Table 4 and assumes that subjects bid in proportion to the number of
delegates from winning the state. Hence the proportional bid for the (4,1,1,1) map is given by
(4(24/7), 24/7, 24/7, 24/7) and for the (2,2,2,1) map by (2(24/7), 2(24/7), 2(24/7), 24/7). The
values for the (3,2,1,1) map can be computed similarly.
We then sort every bid of every subject on the three different maps into one of these three
categories and present this information in Table 5. We do this by taking the subject’s bid value in
every round for a given map and compute its Euclidean distance from the N, B and P points defined
above respectively. The observed bid is assigned to the category to which it is closest in terms of
9
Hence the Nash strategy for the (4,1,1,1) map is bidding 12 while the Behavioral one is bidding 24 on the 4 delegate
state, and zero on all the other three states. Similarly, for the (2,2,2,1) map, our behavioral strategy will consider
situations where the players bid on any two states with 2 delegates, and will use the value 8 given by the Behavioral
column instead of 6 under the Theoretical column.
20
distance. This exercise is repeated for all three maps. The last row labeled “Zero” is an additional
category that computes the distance of the actual bid to the Origin and puts it in this category if it
is closer to the Origin than either of the points N, B or P. Of course actual zero bids were already
recorded in Table 4 and are shown again in parenthesis in Table 5. The much larger number of
zero bids seen here are due to the large number of very small bids, with the frequency of bids less
than or equal to one for each state shown in Table 6.
Table 5. Classifying bid types by their magnitudes (in percentage terms)10
Closest Pattern
4,1,1,1
3,1,1,2
2,2,2,1
Nash
11.2
10.2
17.5
Proportional
(=Everywhere)
19.2
23.0
31.0
Behavioral
53.2
51.0
35.2
18.2 (4.7)
16.5 (6.0)
18.7 (6.0)
(=Minimal)
Zero
Table 6. Bids less than or equal to one by state and map (in percentage terms)
(4,1,1,1)
(3,1,1,2)
(2,2,2,1)
North
11.0
22.8
19.5
East
71.5
35.7
19.8
West
72.3
41.2
26.8
South
73.7
38.3
62.7
The first thing to observe from Table 5 is that it reinforces what we know from Table 4 – the
Electoral College game is a difficult game with very few subjects actually playing the Nash
equilibrium. The behavioral strategy seems to be doing best. At first glance this might seem
counter to what we found in Table 4 since people do not seem to be bidding on minimal winning
10
Note that there are a few cases a bid may be equally close to more than one strategy due to which the
percentages add up to a little over 100 percent for all three maps.
21
sets. This indeed true – subjects seem to be bidding on more than two states, but they seem to be
spending only small amounts (see Table 6) on the states that a not a part of the winning
combination. Hence even though they are not bidding on the minimal set, in terms of Euclidean
distance, point B seems to be the closest. This is driven by two factors – over 70% of the subjects
use a budget of 24 in all three maps. Moreover, subjects also seem to intuitively understand the
importance of a state and take this into account in their bidding behavior. Similar behavior is
observed in the Hex game of Deck et al. (2016) both when they consider a 22 map: subject’s bids
are proportional to the number of winning paths through a region. Interestingly, they find that such
a behavior is also demonstrated in a 44 Hex map for which even the optimal theoretical solution
is not known.
Note also that there is a significant number of small bids as shown by the Zero row. For this we
see a significant increase over time in the frequency of bids that we classify as the zero strategy,
with such bids occurring 15.4% of the time in the first half of the periods, and 22.5% of the time
in the second half though the majority of this is driven by the (2,2,2,1) map. One possible reason
could be that given the difficult nature of the game, subjects may just be giving up in the latter half
of the game, because subjects have to pay their bids even if they do not win. This is particularly
likely to the be the case for (2,2,2,1) map (with its three possible minimal winning sets that are all
identical), since in the other maps we do not see a change in zero bids between early and late
periods. Another pattern that is also clear here is that rather than splitting their bids equally over
the states for which a positive bid is placed, in (3,1,1,2) subjects appear to be allocating money
among the states for which a positive bid is placed in proportion to the weight of the state suggested
by the Behavioral column in Table 2.11
Finally, another question that we investigate is whether is if player strategies are influenced by the
previous map they played on. However, we find no such ordering effects in our data. Players do
appear to treat each map as a separate game with little, if any, carryover from the previous maps.
5. Discussion
In this paper we develop a model of Electoral College competition based on the lottery success
function without an explicit budget making it different from work on stochastic Colonel Blotto
games. Since the different states have different weights, our paper is closer to the literature on
asymmetric Blotto games. First, it is worth noting that unlike Duffy and Matros, our paper is able
to solve the problem for an Electoral College competition with any number of votes. Strategically,
it takes into account the contest nature of the Electoral College as well as the complementarities
by accounting the importance of each state in being able to provide a win. The experiment provides
11
For (2,2,2,1) this distinction is not relevant because the all states in a minimal winning set have an equal number
of delegates.
22
valuable insights about Electoral College behavior since out laboratory set up does not have the
deficiencies present in the empirical data on campaign spending.
Our paper also relates very closely to the theoretical work of Kovenock et al. (2015) and the
experimental work of Deck et al. (2016) on contests with complementarities. The key difference
being that in the former the set of winning combinations differ for the different players while they
are identical for both players. Comparing these two papers provides several interesting stylized
facts and suggests possible directions for future research. First it is evident that contests with
complementarities are difficult for experimental subjects and they rarely behave according to
theory. Instead, as with many complex problems, they turn the simultaneous decision problem into
a sequential decision problem by first choosing the budget and then deciding how to allocate it. In
setting the budget there seems to be a mental accounting rule in place where subjects chose to
spend about half the value of the prize as their budget. Another fact that seems to be common to
both types of contests with complementarities is that subjects seems to be following a trial and
error type heuristic in deciding how to allocate resources over states. They stick to a strategy
switching only when they lose. Both of these raise questions about what drives the strategic
behavior of subjects at a more fundamental level. For instance when do subjects start using a
sequential procedure as a way to make decisions in a simultaneous move game? How does the
mental accounting strategy depend on the value of the prize or the number of participants in a
contest?
6. References
J. Banzahf. Weighted voting doesn’t work: A mathematical analysis. Rutgers Law Review
vol.19, pp. 317-343 (1965).
Emile Borel. La theorie du jeu les equations integrales a noyau symetrique. Comptes Rendus del
Academie vol 173, pp. 1304–1308 (1921)
Emile Borel, and J. Ville. Application de la theorie des probabilités aux jeux de hasard.
Gauthier–Villars, Paris. In: Borel, E., Cheron, A. (Eds.), Theorie mathematique du bridge a la
porteé de tous. Éditions Jacques Gabay, Paris, 1991 (1938)
Emmanuel Dechenaux, Dan Kovenock, and Roman Sheremeta. A survey of experimental
research on contests, all-pay auctions and tournaments. All-Pay Auctions and Tournaments
(September 28, 2012) Available at SSRN: http://ssrn.com/abstract=2154022 (2012)
23
Derek J. Clark and Kai A. Konrad. Contests with multi-tasking. Scandinavian Journal of
Economics vol. 109, pp. 303-319 (2006)
Cary Deck, Sudipta Sarangi, and Matt Wiser. An experimental investigation of simultaneous
multi-battle contests with strategic complementarities. Journal of Economic Psychology (2016)
Cary Deck and Roman M. Sheremeta. Fight or flight? Defending against sequential attacks in the
game of siege. Journal of Conflict Resolution (2012)
John Duffy and Alexander Matros. Stochastic asymmetric Blotto games: Some new results.
Economics Letters vol. 134(C), pp. 4-8 (2015).
M. Lake. A New Campaign Resource Allocation Model. In S.J. Brams, A. Schotter and G.
Schwodiaeur (eds.). Applied Game Theory. Wurzburg, West Germany: Physica-Verlag. (1979)
Lawrence Friedman. Game theory models in the allocation of advertising expenditures.
Operations Research vol. 6, pp. 699-709 (1958)
Gimpel, J. G., Kaufmann, K. M. and Pearson-Merkowitz, S. Battleground States versus Blackout
States: The Behavioral Implications of Modern Presidential Campaigns. Journal of Politics, 69:
786–797. (2007)
Russell Golman and Scott E. Page. General Blotto: Games of allocative strategic mismatch.
Public Choice vol. 138, pp. 279-299 (2009)
Oliver Alfred Gross. The symmetric Blotto game. Rand Corporation Memorandum, RM-424
(1950).
Oliver Alfred Gross and R. A. Wagner. A continuous Colonel Blotto game. Rand Corporation
Memorandum, RM-408 (1950)
Kai A. Konrad. Strategy and Dynamics in Contests. Oxford University Press, (2009)
Dan Kovenock and Brian Roberson. Conflicts with multiple battlefield. CESifo Working Paper,
3165 (2010)
Dan Kovenock, Sudipta Sarangi, and Matt Wiser. All-Pay 2 X 2 Hex: A multibattle contest with
complementarities. LSU Working Paper, 2012-06.
Anne O. Krueger. The political economy of the rent-seeking society. The American Economic
Review vol. 64, no. 3 pp. 291-303 (1974)
Dmitriy Kvasov. Contests with limited resources. Journal of Economic Theory vol. 136 pp. 738748 (2007)
24
Jean-Francois Laslier. How two-party competition treats minorities. Review of Economic Design
vol. 7, pp. 297-307 (2002)
Jean-Francois Laslier and Nathalie Picard. Distributive politics and electoral competition.
Journal of Economic Theory vol. 103, pp. 106-130 (2002)
Brian Roberson. The Colonel Blotto game. Economic Theory vol. 29, pp. 1-24 (2006)
Shaw, D. R. (1999). The methods behind the madness: Presidential electoral college strategies,
1988-1996. The Journal of Politics, 61(4), 893-913.
James M Snyder. Election goals and the allocation of campaign resources. Econometrica vol. 57,
pp. 637-660 (1980)
Balázs Szentes and Robert W. Rosenthal. Three-object two-bidder simultaneous auctions:
Chopsticks and tetrahedra. Games and Economic Behavior vol. 44, no. 1 pp. 114-133 (2003a)
Balázs Szentes and Robert W. Rosenthal. Beyond chopsticks: Symmetric equilibria in majority
auction games. Games and Economic Behavior vol. 45, no. 2 pp. 278-295 (2003b)
Gordon Tullock. Efficient rent seeking. Towards a Theory of the Rent Seeking Society, edited by
Gordon Tullock, Robert D. Tollison, James M. Buchanan, Texas A&M University Press, 3-15
(1980)
Appendix
Subject Instructions
This is an experiment about the economics of decision making. You will be paid in cash at the end of the
experiment based upon your decisions, so it is important that you understand the directions completely.
Therefore, if you have a question at any point, please raise your hand and someone will assist you.
Otherwise we ask that you do not talk or communicate in any other way with anyone else. If you do,
you may be asked to leave the experiment and will forfeit any payment.
The experiment will proceed in three parts. You will receive the directions for part 2 after part 1 is
completed and for part 3 after part 2 is completed. What happens in part 2 does not depend on what
happens in part 1, and what happens in part 3 does not depend on what happens in part 1 or 2. But
whatever money you earn in part 1 will carry over to part 2 and whatever you earn in part 2 will carry
over to part 3.
Each part contains a series of decision tasks that require you to make choices. For each decision task,
you will be randomly matched with another participant in the lab. None of the participants will ever
learn the identity of the person they are matched with on any particular round.
In each task you have the opportunity to earn Lab Dollars. At the end of the experiment your Lab
Dollars will be converted into US dollars at the rate $25 Lab Dollars = $US 1.
25
You have been randomly assigned to be either a “Green” or a “Yellow” participant. You will keep this
color throughout the experiment. For every task you will be randomly matched with a person who has
been assigned the other color. Your color is indicated in a box on the lower part of your screen. This
portion of your screen also shows you your earnings on a task once it is completed and your cumulative
earnings in the experiment. You will start off with an earning balance of $160.
You and the person with whom you are matched are facing each other in an election. In the center of
your screen you can see a map with four regions: North, South, East, and West. Each region has an
associated number of delegates as will be shown on the map with a number beside a “D”. If you win a
region you claim all of the delegates from that region. The person who wins a majority of the delegates
on the entire map wins the election and receives a payment of $48. Since there are always a total of 7
delegates, anyone who wins 4 or more delegates wins the election. This map will not change during this
part of the experiment.
You and the person you are matched up with have to decide how much to spend campaigning in each of
the four regions. Any amount you spend on a region is deducted from your earnings regardless of
whether or not you win the region. Since you have to pay what you spend, the sum of your campaign
spending in the four regions cannot exceed the payment you would receive if you win, $48.
Claiming the delegate in a region works in the following way. The chance that you claim a region is
proportional to how much you spend campaigning in the region relative to the total amount spent for
that region. For example, suppose that Yellow spends $6 for the North and Green spends $2 for the
North then the chance that Yellow would claim North is 6/(6+2) = 6/8 = 75%. The chance that Green
would claim North is 2/(6+2) = 2/8 = 25%.
As another example, suppose that Yellow spends $0 for the North and Green spends $0.25 for the North
then the chance that Yellow would claim North is 0/(0+0.25) = 0%. The chance that Green would claim
North is 0.25/(0+0.25) = 100%.
If both parties spend $0 then each would claim the region with a 50% chance.
You and the person you are matched with will both privately and simultaneously determine your
spending for all four regions at one time. The computer will then determine who claims each region
based upon the probabilities associated with the spending. Each region will turn Yellow or Green to
indicate who claimed the delegates in that region.
Summary Announcements
1. You receive the prize of 48 only if the total number of delegates in the set of regions you claim is
at least 4. It does not matter how many regions you claim, only how many delegates you have.
2. The chance you claim a region is equal to the proportion of your bid on the region to the total
amount bid on that region. For example, if you bid 10 and the other person bids 20 then you
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have a 1/3 chance of claiming the region and the other person has a 2/3 chance of claiming the
region.
3. Any amount you bid is automatically deducted from your earnings, regardless of whether or not
you receive the prize of 48. For example, if your four bids sum to 40 and you receive the prize
you will earn 8 because you will earn the prize of 48 – 40 in bids, but if you do not receive the
prize you will earn – 40.
4. The sum of your bids cannot exceed 48.
5. You must enter a numeric bid for each region. If you want to bid 0 for a region you must type in
a 0, you cannot leave it blank. Bids can have up to 2 decimal places.
6. If you press the submit button and it does not disappear, then your bids have not been
submitted. In this case you need to make sure that you have entered a numeric bid for each
region, that each bid is non-negative, and that the sum does not exceed 48.
7. Each round you are randomly matched with someone in the experiment. So there is a chance
that you interact with the same person two times in a row, but it is unlikely.
Any questions before we begin?
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