Informing Defensive Strategy in Cornhole Through Probabilistic Modeling

Abstract

The game of cornhole is structured quite simply: hierarchically from round to game, a game is won by scoring or exceeding 21 points before your opponent. Because each game’s score is the sum of the point differential in each round, to optimize the probability of winning a given game the objective should be to create as large a favorable point differential as possible in each round. With no obvious method of defense, the game has an inherently offensive-centric nature. To score points, players primarily aim to make the bag into the hole on the board, yielding three points, whilst preserving a point if the bag misses the hole but remains on the board. However, this single faceted approach is too narrow. Because both players throw to the same board alternatingly, bags situated in front of the hole can act as barricades to the opponent’s subsequent throws, affording players the opportunity to establish defensive positioning. In this paper, we propose a new strategy to the game, examining the defensive potentiality of each given throw. Through the use of OpenCV, the spatial coordinates of each bag’s position on the board can be retrieved, and, in conjunction with a play-by-play log, we use this data to transform the cornhole board into a probabilistic grid in order to assess the value of a given throw with consideration given to the bag’s defensive impact. Applying a simulation accounting for human error in throws, we show that abiding by the deduced strategy creates a significantly larger point differential (in favor of the player abiding by the strategy) and accordingly increases a given player’s win probability.

 
  1. Introduction

Across all interaction sports, most fundamental to any calculus of defensive consequence is some valuation of spatial significance. Likewise, in cornhole, the base of our defensive evaluation model is some metric by which to evaluate area of effect for the bag. A multitude of papers have elicited various methods by which to estimate space domination in sport, with the chief denominating method being an implementation of a Voronoi tessellation. Treating an entity as a point, Voronoi tessellations declare all regions in a plane closest to a given point space owned by that point. Most recently, Javier Fernandez et al. advanced past a baseline Euclidean distance model and worked to incorporate entity externals in order to represent the continuous rather than discrete nature of space domination.

However, such methodology cannot be adapted for target sports (such as golf, billiards, cornhole, etc.). In target sports, entities are sedentary; therefore, the indication of space domination can’t be expressed by a concept of “space ownership”. Rather, in target sports, a metric for spatial dominance must be a factored product of some prediction of influence of future shots and the variation in the value of these influenced shots a determination of the defensive value of the controlled area where proximity to the target is not the main factor of relevance for positional value. Thus, goal of this paper is to develop metric of spatial dominance and influence in target sports, using comprehensive probabilistic modeling with cornhole as a prototype.

2. Rules and Terminology of Cornhole

  • Three Categorizations for Shot Outcome
    • Cornholes (3 points)
      • Shots making it into the hole
    • Woodys (1 points)
      • Shots landing on the board but not in the hole
    • Foul Shots (0 points)
      • Shots landing off the board or are discounted for violation
  • Types of Shots
    • Slide
      • Player attempts to slide the bag from a spot on the board into the hole
    • Airmail
      • Player attempts to land the bag directly into the hole without contacting the board

3. Determining Defensive Valuations

In order to determine the value of a given shot’s defensive positioning, we must establish a model evaluating the likely domain of influence of the bag by a generalized position. This task is twofold; we must first devise a method by which to assign the bag a generalized position, and secondly, we must deduce the likelihood each generalized region influences a given shot.


3.1. Defensive Spatial Influence

To assign the bag a generalized position on the board, we sectionalize the board, transforming it into a 16×32 tile grid. We then determine the area each tile occupies and assign this as the area domination variable of the tile.

Advancing to a determination of regional influence, we first limit our sample of shots to those taken on an open board (0 bags on the board) in order to analyze each bag’s defensive influence in isolation before accounting for context. To produce such a situation, we use a comprehensive sample of shots thrown on an open board, and using a divisive analysis hierarchical clustering model, we can assign each of these shots into a shot path cluster[1]. We then determine the tiles each trajectory cluster intersects and assign a density value according to the number of shots in the cluster.

3.2.  Positional Valuation

Furthermore, using the shot trajectory clusters as covariates, we construct a binomial logistic regression model setting the response variable as shot outcome—cornhole or woody (omitted foul bags from this regression due to insufficient sample)—we can determine probability of a make/miss and therefore can identify an expected value of the shots predicted to intersect with weights given according to cluster density. Designating this figure as “EVIS” (Expected Value of Intersecting Shots), we subtract this value by value by 1 to find points saved (if blocked, a woody still yields 1 point; thus, points saved is value of a woody subtracted from the mean expected value of intersecting shots) and multiplying it by the intersection probability, we can determine a locational value ranging from 0 to 1.5, representing likely points saved per subsequent opponent shot attempt.

3.3.  Contextualizing Positioning

Because shots are thrown alternatingly on the same board, the necessary extension of this calculation is a consideration of overlapping samples of influence. That is, for an accurate representation of defensive value, each bag’s positional value needs to be considered relatively to all shots preceding it, thus reflective of defensive value added rather than purely isolated defensive value.

In order to represent defensive value added, it was determined from the aforementioned sample what percentage of clustered shot trajectories intersected both the area of the given bag and a bag thrown prior to the given bag (within the same round). This sample of shots, deemed to be influenced by both bags, was removed from the sample of the given shot, thus creating a value-added model of determination.

3.4.  Knock-In Probability

A significant factor when determining positional value in cornhole is the probability that a given bag on the board will be knocked into the hole by a subsequent shot. This occurrence is called a “knock-in”. Treating the previously calculated holistic defensive value added as the principal perceptible determinant in “knock-in” probability, where the covariate is the defensive value added of a given shot[2] and the response variable is whether or not a bag is knocked into the hole on the subsequent shot.

Figure 1

Using this curve, we can establish  as subsequent shot knock-in probability for shot n, and thus cumulative shot knock-in probability for shot can be represented as:

4.  Knock-In Probability

Now that we have determined defensive impact of a given bag relative to context, it is necessary to expand the model in order to determine net defensive impact on all shots subsequent to the given shot. In this analysis, only the outcome of some singular shot is known, and the events occurring prior to the shot; thus, it is not known which events will occur in the future and at what time the events will occur. Therefore, net effect can most accurately be determined by determining all possible outcomes and deducing the probability of such outcomes. In order to model the net effect of a given defensive shot in cornhole, we can use a decision tree consisting of chance nodes, assigning the “test” value as the result of the shot (i.e. make or miss).

Figure 2: Decision Tree Diagram

This tree, when adapted to a given circumstance, initializes from a given missed shot at any point in a round, designated as initial interval node , where each subsequent shot would be interval node +1, The model predicts airmail attempts at instances where the expected points yielded from an airmail attempt (model for expected points from an airmail attempt explained in appendix x) exceeds that of a slide.

In order to calculate values for each node of , we assume the defensive points added of each subsequent shot as replacement level. In order to determine the replacement level of a given shot, we construct a Bayesian ordered regression proportional odds model, where the observation is the number of bags on the board and the response variable is the likelihood of each shot outcome (cornhole, woody, foul).

We can thus express total expected points per node as:

Where  is the cumulative probability of a cornhole in a given node (defined in appendix),  is the cumulative probability of a foul shot in a given node

Plotting the output of (2) as the dependent variable against the independent variable of defensive points added for node , we retrieve Figure 2, fit with a loess function.

Figure 3: Defensive Points Added vs Expected Points per Node

5.  Situational Example

To provide an example analysis, we look back to the 2017 ACL (American Cornhole League) Championship of Bags third place match between Jordan Langworthy (pink bags) and James Baldwin (red bags). Considered one of the most exciting cornhole games in history, we look towards the final round, in which Jordan Langworthy overcame a one-point deficit in order to win the game. The sequence of shots and their corresponding outcomes are represented below in Table 1.

Shot Number (Within Round)Langworthy ScoreBaldwin ScoreAttempt Type
110Slide
231Slide
341Slide
444Airmail
554Slide
675Slide
785Slide
886Slide
Table 1: Round Summary

We consider the first shot, expressed below in Figure 3, and its associated deterministic constants identified previously.

Figure 4: Langworthy Shot1 Plot (Color Gradient Representative of EVIS, Arrow Size Representative of Cluster Density)
Table 2: Deterministic Coefficients of Langworthy Shot1

Logically, a significant implication of the bag in the given position is a high likelihood of being knocked-in by a subsequent throw. With the bag in the given position, Langworthy functionally limits Baldwin to three potential strategies: attempt for a cornhole entering the hole on the right, attempt to push both his and Langworthy’s bag into the hole, or throw his bag in front of Langworthy’s and settle for a “pile-up”. However, to examine the likely implications of this throw, we use (2) for net effect to predict the parameters in Table 3.

Table 3: Resultant Table of Langworthy Shot1

Using the equation given in (3), we expect Jordan Langworthy to score 7.47 points, and James Baldwin to score 6.91 points, thus returning the projected net differential throughout the round to be 0.56 points in favor of Jordan Langworthy. Four shots later, following the series delineated in Table 1, we arrive at the situation in Figure 4 following Jordan Langworthy’s third throw (initial interval node becomes shot 189).

Figure 5: Langworthy Shot3 Plot (Color Gradient Representative of EVIS, Arrow Size Representative of Cluster Density)
Table 4: Deterministic Coefficients of Langworthy Shot3
Table 5: Resultant Table of Langworthy Shot3

Using the equation given in (3), we expect Jordan Langworthy to score 3.12 points across this series of shots (including initial), and James Baldwin to score 2.73 points across the same series, thus returning the projected net differential across the four shots to be 0.39 points in favor of Jordan Langworthy.

6.  Conclusion

In this paper, we have expressed a viable method by which to evaluate defensive positioning in cornhole and use it to forecast outcomes of given rounds. We have shown how to express defensive positioning in terms of likelihood of shot influence and have likewise shown how to determine a weighted point value for given spaces on the cornhole board. Finally, we have shown a positive relationship between defensive points added and expected net points of a given node, both in a focused and expansive scale.

References

[1] COBS 2017  James Baldwin vs Jordan Langworthy  3rd Place Singles. (n.d.). Retrieved from https://www.youtube.com/watch?v=vzZww8kSIko&t=64s

[2]Fernandez, J., Barcelona, F. C., & Bornn, L. (2018). Wide Open Spaces: A statistical technique for measuring space creation in professional soccer, 19.

[3]Official Cornhole Rules by the ACO – American Cornhole Organization. (n.d.). Retrieved December 16, 2019, from https://americancornhole.com/rules/

Appendix

Models

Expected Points of Airmail Attempts:

The model for expected points of airmail attempts is simply represented by:

There was no substantial correlation identified between board conditions and airmail success rate/yielded value; thus, as airmail conversion probability is largely stochastic, a baseline sum of points over total attempts expected yield model is valid.

Expected Points Model Expanded


[1] Validated using sum of squares method

[2] We are able to treat defensive value added as the covariate in this calculation because there are two principal factors which conceivably influence knock-in probability: number of bags on the board and the given bag’s susceptibility to be knocked-in. Both of these values are represented in the defensive points added model, where congestion is inherently featured as it is a points-added model and thus if there is a bag in front it will have a lower defensive points expected, and a bag’s susceptibility to be knocked in is modeled by EVIS and density of the cluster

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