Winning Probability

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.)

WinningWithSalRatioEq1

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

where

  • 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.

Density Curves for Performance for Different Saliency Values

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.

Two density curves with saliency ratio = 2, and power difference = 1

Two Density Curves for Participants with Power Difference = 1 Standard Deviation

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.

power difference = 0, saliency ratio = 2, and winning probability = .5

With no power difference

power difference = 0.5, saliency ratio = 2, and winning probability = .6915

With power difference = .5 standard deviation

power difference = 1, saliency ratio = 2, and winning probability = .8413

Two Density Curves with power difference = 1

power difference = 2, saliency ratio = 2, and winning probability = .97725

Two Density Curves with power difference = 2

power difference = 4, saliency ratio = 2, and winning probability = .999968

Two Density Curves with power difference = 4

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.

Winning probability = .6915 with saliency ratio = 1, and power difference = 1

Winning Probability With Saliency Ratio = 1 (for fixed power distance = 1)

Winning probability = .8413 with saliency ratio =2, and power difference = 1

Winning Probability With Saliency Ratio = 2 (for fixed power distance = 1)

Winning probability = .9332 with saliency ratio =3, and power difference = 1

Winning Probability With Saliency Ratio = 3 (for fixed power distance = 1)

We normally do not know exactly which saliency ratio is actually operating, other than that this ratio is likely to change as we consider different types of competitions. The saliency ratio is likely to be greater for more serious competitions, such as war, than more trivial competitions, such as sports games. So a general model of winning probability must incorporate two parameters–the power difference and the saliency ratio. Such an interactive model is shown in the following density graph for bilateral competitions. If you have installed CDF Player, you will be able to see the interactive graph. By moving the two sliders, you can see how the two parameters affect the winning probability and the associated density curves. For a fixed value of power distance, the greater the saliency ratio, the greater the winning probability of the superior participant; for a fixed value of saliency ratio, the greater the power distance, the greater the winning probability. The combined effects of these two parameters are interactive.

Now, the winning probability in a bilateral competition can be derived, rather easily, given the normality assumption about the error terms.

Calculating the Winning Probability

It is important to note the following:

  • the density curves shown above are all normal density functions,
  • therefore, they are all symmetrical,
  • two density curves are the same except for the location of the center point (the mean for each distribution)
  • the dissecting point is located at the middle of the two means
  • the probability of winning for the superior participant is given by the area above the dissecting point of the right density curve, which is the same as the area below the dissecting point for the density curve on the left

The area of the density curve below the dissecting point on the left can be expressed in terms of the normal cumulative probability function, by simply substituting the limits of integration ( u ) in the following normal cumulative distribution function (CDF) with u = 1/2*δ*(α/β) where δ is the power difference between the two participants, measured in standard deviation units, α/β is the saliency ratio:

Formula_WinningCDF

The above formula makes the loosely stated propositions into an exact function: as you can surmise, as the magnitude of δ increases, the winning probability increases, and as the value of saliency ratio, α/β , increases, so does the winning probability. But the connection among these parameters are complex and interactive. That is, the impact of these parameters are interactive, and not additive.

The CDF values for the normal density function is usually available as an appendix for most books on statistics. When the saliency ratio is set at 2 (α/β = 2), u = δ, and the winning probability of a superior participant with δ-power difference from the weaker participant can be read off from the standard CDF table. But with the wide availability of computers and computing programs, the exact calculation of the associated probability is at our fingertips. In the following we show the relationship between the density curve, the CDF curve, and their connections to power-distance and the saliency ratio by way of interactive probability graphs.

PDF, CDF, and Power Difference: a graphic illustration

Showing the connection among power difference, pdf and cdf

The above graphs are PDF and CDF graphs from a unit normal distribution. The PDF (probability density function, whose value is given by the height of the curve) is the derivative of the CDF (cumulative distribution function), which is simply the rate of change of the CDF curve.

The shaded area in the probability density function represents the winning probability graphically, while the height of the intersecting point on the cumulative distribution function shows the winning probability for the corresponding x-value on the horizontal axis. The values on the horizontal axis represent the power-distances between the two participants. Here it is shown that for a power-difference of 1 standard unit, the winning probability of the stronger participant over the weaker one is .841. The unit-normal distribution is appropriate only when the saliency ratio is assumed to be 2. When the saliency ratio gets bigger, the spread of the normal density curve becomes narraower, and the incline of the cumulative distribution function becomes steeper. As the saliency ratio decreases, the opposite is true: the spread of the density curve becomes wider, and incline of the cumulative function becomes less steep and more linear.

One general pattern remains: the relationship between the power-difference and the winning probability is monotone increasing, but not completely linear. This non-linearity increases as we move near 1 or 0 probability, and as the value of saliency ratio increases.

Th entries in the Cumulative Normal Distribution table included in many statistics books are no more than a limited selection of some values from this connection shown in the above graphs. For those without access to appropriate computer programs, this table can be used to get the winning probability values for any value of power difference (in standard units) with any assumed value of saliency ratio, by using the following transformation:
z = 1/2*δ*(α/β), where δ is the power-difference and (α/β) is the assumed saliency ratio.

For a full exposition of the underlying statistical foundation, see Jae-On Kim, Sanghag Kim, and Jin Wang, “Index of National Power: How to Assess the Basic National Capacity of a Nation,” Korean Journal of Sociology, 2013, Vol. 47(6):83-140.