In this section, we introduce a new statistical model with which to connect winning (and losing) probabilities to the underlying capacities of participants in any competition. The fundamental assumption is that an index of power (capacity, power-potential, strength, whatever the name may be) has meaning only if it has some systematic and credible connection with the winning probabilities in a competition.
Since most of us are unfamiliar with probabilities, in general, and the probability of winning or losing in a competition, in particular, it is useful to start with some clear frame of reference. So let us start with a simple pictorial presentation of two density functions, each of which represents a long-term (theoretical) distribution of performances of a team. In this particular example, we have contrived the two teams to have one-standard deviation difference in their underlying capacity. In this graph, you see two partially overlapping normal density functions. (You might wonder, why we suddenly switch from probability to density. But let us ignore the technical difference for now and assume that the relative height of the normal curve indicates the differences in probability for a given performance level which is given by the horizontal x-axis.)
The first feature to note is the pattern of the density functions (the shape of the normal curves): it peaks at the center–the corresponding x-value is known as expected value–and it dwindles as the value of x moves away from this center. One feature that is not clearly visible is that each curve has long tails at both sides.
The second feature to note is the difference in the expected values, or the positions of the center of the two density functions. These expected values are direct consequences of the differences in the underlying capacities of the two teams. It shows that the stronger team is likely to perform better in the long run, and the greater the difference in the underlying capacities between the two teams, the greater the difference in the expected values. That is why we often try to gauge the strength or power of a team and the relative power differences of the two competing teams.
The third feature to note is that the two density functions overlap, meaning that an upset is possible: a weaker team’s performance at a particular contest may be better than that of the stronger team. Given the pattern of the normal density functions, there will be some overlap almost all the time, and, therefore, the outcome is never a certainty. That is why they play the game!
Finally, the fourth point to note is that the only thing we can get out of the power difference is the probability of winning or losing, or in technical terms, an expected value from a theoretical distribution that imitates limitless bilateral competitions between two teams with a particular difference in their underlying capacity.
In order to make these observations into a more formal setting, we introduce a simple model of bilateral competition in which the relationship between the capacity (or power) of each participant or competitor and the winning probability of each competitor is clearly spelled out. We start with the following simple formula which links the underlying capacity to a performance:
pij = α Cj + β ei
- pij stands for j’s competitive performance in a particular (i) competition
- C stands for the true (underlying) capacity
- e stands for random chance component
- α stands for the coefficient for the true component
- β stands for the coefficient for the error component
Without losing generality, the following assumptions are imposed:
- var(C) = var(e)= 1.0
- C2 + e2= 1.0
- e comes from a normal, independent, and identical distribution
The density for performance outcome will then follow the normal density function, with the mean at C and the spread of the distribution determined by the saliency ratio ( α/β). The following graph shows four density curves, with different values of the saliency ratio, ranging from 0.5 to 3. As the value of saliency increases, the role of random elements decreases, making the spread of the density curve narrower.
Note that the height of these density graphs can be greater than 1, clearly showing that density is not a probability. But the relative height is clearly linked to probabilities, to be explained later.
Now we are ready to introduce a winning probability model in a bilateral competition. When two participants compete, we do not know exactly how each participant is going to perform at a given contest, because of the random component. Given the model of performance shown above, however, it is possible to obtain the winning probability for each participant. It is important to note that the “probability to win” here is given by the limit approached by the relative frequency of winning when the number of repeated contests approaches infinity.
Winning and Losing in a Bilateral Competition
Starting with the simple linear model introduced in the previous section, we now introduce a winning probability model for a bilateral competition. To meet the fundamental axioms of a probability model, the proposed model must show the following relationship between the index of power and the winning probability:
- if CA = CB, then winning probability of A or B = 1/2; that is, WP(A) = WP(B) = 1/2
- if CA > CB, then winning probability of A over B > the winning probability of B over A; that is, WP(A)>WP(B)
- WP(A) + WP(B) = 1.0
- WP(A) + LP(A) = 1 = WP(B) + LP(B)
- winning or losing probability of A or B >= 0
Such a probability model can be constructed rather easily by examining the performance density curves of two participants. To fix the idea, consider a case in which two participants A and B with a power difference of one standard deviation compete in a contest. As we noted, we do not know the outcome in a particular contest, but we know the expected probability.
In the above density graph, the shaded area for the density function for A (the blue curve on the right) represents the winning probability of A over B, and the un-shaded area of the blue density curve represents the losing probability of A. Likewise, the un-shaded area of the density curve for B on the left represents the losing probability of B, which is the same as the winning probability of A, while the shaded area for that density curve represents the winning probability of B over A, which is equal to (1 – WP(A)). The assumption of normality of the error component makes the calculation of these probabilities rather simple. In this case, the winning probability of A over B is .841345, with a power difference of 1 standard deviation (with assumptions about random components).
In the following, we show several graphs with varying power difference, ranging from 0 to 4 standard deviations. Note as the power difference increases, so does the winning probability of A over B.
Note as the power difference increases from 0, .5, 1, 2, to 4, the probability of winning increases from .5, .691462, .841345, .97725, to .999968. The relationship between the power difference and the winning probability is monotone increasing, but not linear. This type of non-linearity is expected given the bounded nature of probability between 0 and 1.
Saliency ratio also affects the winning probability. Given a fixed power difference between the two participants, the higher the saliency ratio, the higher the winning probability. With a fixed power difference of 1 standard deviation, the following three graphs show the increasing probability from .6915, .8413, to .9332 as the saliency ratio changes from 1, 2, to 3.