Pub points — should I go to the Pub after a win, draw or loss?
I play in a weekly league game with friends, where our individual rankings are based on the overall average points per game. Here’s how the scoring system works:
- Win: 3 points
- Draw: 1 point
- Loss: 0 points
- Pub Attendance: 0.5 points
Since I can’t make it to the pub after every game, the question arises: Should I prioritise going to the pub after a win, draw, or loss? Below are responses from different AI models, each with their own take on the optimal strategy.
My prompt for each LLM
“I play in a weekly league game with friends. Individually we each get 3 pts for a win. 1 pt for a draw. 0 pts for a loss. We also receive half a point if we go to the pub after the game. The league is sorted by the highest to lowest points per game. This is your total points divided by number of games played. If I cannot make it to the pub for every single game. Is it better to go to the pub after losses, draws or wins and why?”
Tldr (some of these models thought about this problem for a long time so here is a summary):
ChatGPT-o1
Optimal Strategy for Pub Attendance to Maximize Points Per Game
To determine the most advantageous strategy for attending the pub after your league games, we need to analyze how the additional half-point influences your overall points per game based on different game outcomes: wins, draws, and losses.
Understanding the Points System
Your league awards points as follows:
- Win: 3 points
- Draw: 1 point
- Loss: 0 points
- Pub Attendance: 0.5 points
The league ranking is determined by points per game, calculated as:
Points Per Game=Total PointsNumber of Games Played\text{Points Per Game} = \frac{\text{Total Points}}{\text{Number of Games Played}} Points Per Game=Number of Games PlayedTotal Points
Evaluating Pub Attendance After Different Outcomes
Let’s consider the impact of attending the pub after each type of game outcome.
After a Win:
- Base Points: 3
- With Pub: 3+0.5=3.53 + 0.5 = 3.53+0.5=3.5 points
After a Draw:
- Base Points: 1
- With Pub: 1+0.5=1.51 + 0.5 = 1.51+0.5=1.5 points
After a Loss:
- Base Points: 0
- With Pub: 0+0.5=0.50 + 0.5 = 0.50+0.5=0.5 points
Maximizing Points Per Game
To maximize your points per game, you should aim to maximize the total points accumulated. Adding 0.5 points after a win provides the highest incremental benefit (from 3 to 3.5 points) compared to adding the same 0.5 points after a draw or a loss.
Mathematically, attaching the bonus to higher base points yields a greater contribution to the numerator in the points per game formula:
3.51>1.51>0.51\frac{3.5}{1} > \frac{1.5}{1} > \frac{0.5}{1} 13.5>11.5>10.5
Thus, prioritizing pub attendance after wins enhances your average points per game more effectively than after draws or losses.
Conclusion
It is most beneficial to attend the pub after wins. This strategy maximizes the additional points earned per game, thereby improving your standing in the league rankings.
Gemini 2.0 pro
Here’s a breakdown of the best strategy, thinking step by step:
Step 1: Points per game after a win + pub
If you win and go to the pub, you get 3 points for the win + 0.5 points for the pub = 3.5 points.
Since you played one game, your points per game is 3.5 / 1 = 3.5.
Step 2: Points per game after a draw + pub
If you draw and go to the pub, you get 1 point for the draw + 0.5 points for the pub = 1.5 points.
Since you played one game, your points per game is 1.5 / 1 = 1.5.
Step 3: Points per game after a loss + pub
If you lose and go to the pub, you get 0 points for the loss + 0.5 points for the pub = 0.5 points.
Since you played one game, your points per game is 0.5 / 1 = 0.5.
Step 4: Comparison and Conclusion
Comparing the three scenarios:
- Win + Pub: 3.5 points per game
- Draw + Pub: 1.5 points per game
- Loss + Pub: 0.5 points per game
It is best to go to the pub after a win, as that will give the most points.
ChatGPT o3-mini
Reasoned for 52 seconds
I’m figuring out whether it’s better to go to the pub after losses, draws, or wins since I can’t always make it. The key here is the points per game ratio, which is total points divided by games played.When I lose, I get 0 points, so going to the pub for half a point would boost my total points slightly without affecting my wins. But for draws and wins, going can add more. Hm, I’ll need to analyze the ratios more carefully!
I’m thinking about how attending the pub after a game affects my points per game (PPG) in the league. If I go, I get a bonus of 0.5 points, but is it better to go after losses, draws, or wins?
The bonus is the same, but when I lose, my expected points are lower, so going to the pub after a loss doesn’t boost my total as much as going after a win. Hmm, maybe it’s better to attend after a win or draw to maximize the ratio.
It seems like the pub makes a consistent 0.5 point difference regardless of whether it’s after a win, draw, or loss — but the strategic impact could still differ. After a loss, 0.5 points is a bigger relative percentage increase (since you get 0 points for loss). However, the overall average points per game remains the same because each game counts equally in the ratio. Still, it could be better to go after a loss since the relative improvement is greater compared to going after a win.
If I want to maximize my points per game, the pub bonus of 0.5 affects my total, but its impact depends on when I choose to go. After a loss, turning 0 points into 0.5 has the biggest relative improvement. After a win, that bonus might feel less significant since I’m already scoring high. However, the overall effect is just adding 0.5 to my total, but if I hope to beat others, attending after a loss might give me the best advantage.
It seems like the pub bonus always adds 0.5 points, regardless of whether it’s after a win, draw, or loss — so, in theory, it doesn’t really matter when I attend. However, since the league is based on points per game (PPG), the bonus might be more valuable if I attend after a loss, turning 0 points into 0.5. In contrast, after a win, the bonus has a smaller relative effect on my overall average. If I miss some pub nights, I should likely prioritize attending after losses or draws to maximize my score.
The pub bonus of 0.5 points is a flat addition to your score, regardless of whether you win, draw, or lose. Mathematically, it doesn’t matter when you attend because the extra 0.5 points gets added to the total in all cases. However, the relative impact might be more noticeable after a loss, as you turn 0 points into 0.5 points. Still, the bonus is always the same, so the overall effect on your points per game stands equal. Ultimately, it doesn’t seem to change your ranking unless there are tie-breaker rules.
The pub bonus of +0.5 points adds exactly the same value regardless of whether it’s after a win, draw, or loss. However, its impact on your overall points per game is more significant when you attend after a loss, as it turns 0 points into 0.5. In contrast, a win only increases your score by 0.5 compared to 3 points. So, if I seek to maximize my average, it’s best to go to the pub after a loss to improve my overall ranking.
Each pub visit gives you a flat extra 0.5 points regardless of your game result. In other words, whether you win (3 → 3.5), draw (1 → 1.5), or lose (0 → 0.5), the bonus adds exactly 0.5 points to that game’s score. Since your points per game is just your total points divided by the number of games, the impact on your average is the same no matter which game you get the bonus for.
So, mathematically speaking, it doesn’t matter whether you go to the pub after a win, draw, or loss — the bonus is always +0.5 to your total, and thus your average increases by the same amount.
However, one might note that, in relative terms, turning a loss (0) into 0.5 is a bigger “proportional” improvement than turning a win (3) into 3.5. But since the league ranking is based on the absolute extra points you get (and not on percentages), the overall effect is identical.
Bottom line: It makes no difference which game you attend the pub after — the bonus is the same in every case.
DeepSeek R1 (sorry DeepSeek thought for a long time)
Thoughts
Okay, so I need to figure out whether it’s better to go to the pub after losses, draws, or wins to maximize my points per game in this league. Let me start by understanding the scoring system again.
Each game, I can get 3 points for a win, 1 for a draw, 0 for a loss. Additionally, I get half a point if I go to the pub after the game. The league standings are determined by points per game, which is total points divided by the number of games played. The key here is that if I can’t go to the pub every time I play, I need to strategically choose when to go to maximize my average.
First, let’s break down the possible outcomes and the points associated with them:
Win: 3 points + 0.5 if I go to the pub = 3.5 points
Draw: 1 point + 0.5 if I go to the pub = 1.5 points
Loss: 0 points + 0.5 if I go to the pub = 0.5 points
If I don’t go to the pub, I just get the base points for the result of the game. The question is, if I have a limited number of times I can go to the pub, should I prioritize going after wins, draws, or losses?
Since points per game is the total points divided by the number of games, the strategy would involve maximizing the total points. However, since going to the pub gives an extra 0.5 points regardless of the result, maybe it’s better to go when the extra 0.5 points will have the most impact.
Wait, but points per game is calculated by total points divided by games played. If I go to the pub after certain games, those games will have an extra 0.5 points. So each game where I go to the pub adds 0.5 to the numerator (total points), while the denominator (games played) remains the same. Therefore, each pub visit increases the total points by 0.5. Therefore, to maximize the average, I should allocate the pub visits to the games where the 0.5 points contribute the most to the average.
But wait, actually, all pub visits add 0.5 points, which is a fixed amount. So regardless of whether the game was a win, draw, or loss, adding 0.5 points to that game’s total. However, since the base points for a win, draw, or loss are different, adding 0.5 points might have different proportional impacts.
But since points per game is additive, the average is just (sum of base points + 0.5 * number of pub visits) divided by total games. Therefore, to maximize the average, I need to maximize the total points. Since the number of pub visits is fixed (assuming I can’t go every time, but can choose when to go), then regardless of when I go, each pub visit gives me 0.5 points. Therefore, shouldn’t I just go to the pub as much as possible? But the problem states I cannot go to the pub for every single game. Therefore, I have a limited number of pub visits. Wait, but the problem doesn’t specify how many times I can go. It just says I can’t go every single game. So maybe I can go to the pub for some games, but not all. The question is, should I allocate my pub visits to wins, draws, or losses to maximize the average points per game.
But since each pub visit gives 0.5 points, regardless of the result of the game, then the optimal strategy is to go to the pub as many times as possible. However, the question is not about how many times, but when. Since the user can’t go every time, but wants to know which games to prioritize: after losses, draws, or wins. So perhaps in order to maximize the total points, since each pub visit is worth 0.5, it doesn’t matter which games I go to the pub after, because 0.5 is 0.5. However, the problem might be that if you don’t go to the pub, you get 0.5 less points for that game, but maybe there’s a misunderstanding here.
Wait, actually, let’s clarify: when you go to the pub, you get an extra 0.5 points. If you don’t go, you don’t get that 0.5. So the total points depend on how many times you go to the pub. The more times you go, the higher your total points. But if you have a limited number of pub visits (since you can’t go every time), you need to choose which games to apply those 0.5 points to.
But if each pub visit gives the same 0.5 points regardless of the game’s outcome, then it doesn’t matter which games you choose to go to the pub. For example, suppose you have N games, and you can go to the pub M times. Then your total points would be (sum of base points) + 0.5M, and your average would be (sum + 0.5M)/N. Therefore, to maximize this average, you need to maximize M, i.e., go to the pub as often as possible. But since the problem states you can’t go every time, but the question is, given that you can’t go every time, should you go after losses, draws, or wins? But mathematically, if each pub visit gives the same 0.5, it doesn’t matter which games you choose. Therefore, the answer is that it doesn’t matter; you should go to the pub as many times as you can, regardless of the result.
But this seems counterintuitive. Maybe I’m missing something. Let’s check.
Suppose you have two scenarios:
Scenario 1: You go to the pub after a win and a loss.
Total points: 3 + 0.5 (win) + 0 + 0.5 (loss) = 4 points over 2 games. Average is 2.0.
Scenario 2: You go to the pub after two wins.
Total points: 3 + 0.5 + 3 + 0.5 = 7 points over 2 games. Average is 3.5.
Wait, but this contradicts my earlier conclusion. Wait, wait, no, if you go to the pub after two wins, each win gives 3 + 0.5 = 3.5, so two wins would be 7. But 7 divided by 2 is 3.5.
If you go to the pub after a win and a loss, then it’s 3.5 + 0.5 = 4 over 2 games, average 2.0. So clearly, going to the pub after wins gives a higher average.
But this contradicts the earlier conclusion. So why the difference? Because in the first scenario, the base points are different. So if you allocate your pub visits to higher base point games, you end up with a higher total. Wait, but no — the pub visit adds 0.5 regardless of the base points. Wait in the first scenario, two games: one win with pub, one loss with pub. Total points: 3 + 0.5 + 0 + 0.5 = 4. Average 2.0. Alternatively, two wins, both with pub: 3 + 0.5 each, total 7, average 3.5. If instead, one win with pub and one loss without pub: 3 + 0.5 + 0 = 3.5 over 2 games, average 1.75. So actually, the allocation of pub visits matters because the base points differ. Therefore, adding 0.5 to a higher base point game will give a higher total, but the average is total divided by games, which is the same as (average base points + 0.5 * (number of pub visits)/games). But if you have a fixed number of pub visits, say M, then regardless of where you allocate them, your total points are (sum of base points) + 0.5M. Therefore, the average is (average base points) + 0.5(M/N). Wait, that seems different from the previous example. Wait, perhaps I made a mistake in the previous example.
Wait, let me redo the example. Suppose you have two games. In both scenarios, you go to the pub once.
Scenario 1: You go to the pub after a win. The other game is a loss without pub.
Total points: 3 + 0.5 (win with pub) + 0 (loss without pub) = 3.5. Average is 3.5 / 2 = 1.75.
Scenario 2: You go to the pub after a loss. The other game is a win without pub.
Total points: 3 (win without pub) + 0 + 0.5 (loss with pub) = 3.5. Average is 3.5 / 2 = 1.75.
Wait, so in this case, it doesn’t matter where you allocate the pub visit. The total points are the same. But earlier when I considered two pub visits, I might have confused the number of pub visits.
Wait, hold on, let’s clarify. If you have two games and can go to the pub once:
If you go to a win: 3 + 0.5 + 0 = 3.5
If you go to a loss: 3 + 0 + 0.5 = 3.5
If you go to a draw: 1 + 0.5 + other game (whatever result). Wait, but the other game’s result affects the total.
Wait no, let’s say you have two games: one win and one loss. If you go to the pub after the win, total points are 3.5 + 0 = 3.5. If you go to the pub after the loss, total points are 3 + 0.5 = 3.5. Same average. Similarly, if you have a win and a draw:
Pub after win: 3.5 + 1 = 4.5, average 2.25
Pub after draw: 3 + 1.5 = 4.5, average 2.25
Same result. So regardless of where you allocate the pub visit, the total points added are the same, hence the average is the same. Therefore, my initial thought was correct: the 0.5 points per pub visit are additive regardless of the game result. Therefore, if you have a fixed number of pub visits, it doesn’t matter which games you apply them to; the total points will be the same, hence the average will be the same.
But this contradicts my earlier example where I thought going to the pub after two wins gives a higher average. Wait, no, in that case, if you can go to the pub twice, and you have two wins:
Total points: 3 + 0.5 + 3 + 0.5 = 7, average 3.5
If you have two losses and go to the pub twice:
0 + 0.5 + 0 + 0.5 = 1, average 0.5
But here, the difference is because the base points are different. However, if you have a fixed number of pub visits regardless of the game outcomes, then the average depends on both the base points and the pub points. So if you have a fixed number of pub visits, but the distribution of game outcomes (wins, draws, losses) affects the total base points, then the average will vary based on both your game results and pub visits. However, the question is not about changing the number of wins, but about where to allocate the pub visits given your game results.
Wait, perhaps the confusion arises from whether the game outcomes are fixed or not. The question is: given that you have a certain number of wins, draws, and losses, should you go to the pub after the wins, draws, or losses to maximize your average.
If your game outcomes are fixed, then your total base points are fixed. Adding 0.5 points per pub visit, regardless of which games you add them to, will result in the same total points. Therefore, the average points per game would be (total base points + 0.5 * pub visits) / total games. Since total base points and total games are fixed, the average depends only on the number of pub visits. Therefore, to maximize the average, you should maximize the number of pub visits, regardless of the game outcomes.
But the problem states that you cannot go to the pub every single game. It doesn’t say that you have a fixed number of pub visits. So maybe the question is, given that you have to miss some pub visits, which games should you prioritize attending the pub for: wins, draws, or losses?
But the answer still seems to be that it doesn’t matter because each pub visit gives you 0.5 points regardless. However, this seems off because intuitively, adding 0.5 to a higher base point game would make your total higher. But according to the math, if you have fixed game outcomes, then adding 0.5 to any game just adds 0.5 to the total, regardless of the game’s result. Therefore, the total points added are the same. Therefore, the average is the same.
Wait, let’s take an example. Suppose you have 1 win, 1 draw, 1 loss. You can go to the pub twice.
Option 1: Go to pub after win and draw.
Total points: 3 + 0.5 + 1 + 0.5 + 0 = 5. Average: 5/3 ≈1.666…
Option 2: Go to pub after loss and draw.
Total points:3 +1 +0.5 +0 +0.5=5. Same average.
Option 3: Go to pub after win and loss.
Total points:3 +0.5 +1 +0 +0.5=5. Same average.
So in all cases, the total points are the same, hence the average is the same. Therefore, it doesn’t matter where you allocate the pub visits; the average remains the same. Therefore, the answer is that it doesn’t matter; you should go to the pub as often as possible, regardless of the game’s outcome.
But this contradicts my initial example where two pub visits after two wins gave a higher average. Wait, no, in that example, the total games were two wins, so base points are 6, plus 1 point for two pub visits: total 7, average 3.5. If instead, you had two losses and went to the pub twice, base points 0 plus 1, total 1, average 0.5. But this is because the base points are different. However, the key point is that if the game results are fixed, then the base points are fixed, and adding pub points anywhere adds the same total. Therefore, in that case, it doesn’t matter where you add the pub points. But if you can choose to go to the pub more often when you have better results, then you could influence the average.
Wait, but the problem states that you cannot go to the pub every game. It doesn’t say that you have a fixed number of times you can go. So perhaps the question is: given that you will have some games where you can’t go to the pub, should you skip going to the pub after certain results to maximize your average.
But if that’s the case, then skipping pub visits after certain results would decrease your total points. Therefore, to maximize your average, you should minimize the number of skipped pub visits, i.e., go to the pub as often as possible, regardless of the game’s result.
However, the problem might be that you have to choose in advance to not go to the pub after certain types of games. For example, if you know you will have a certain number of games, and you can only go to the pub a certain number of times, should you allocate those pub visits to games where you win, draw, or lose.
But according to the math, since each pub visit gives 0.5 points regardless of the game’s outcome, you should go to as many pubs as possible. However, if you have to choose, for example, to skip the pub after losses or after wins, the impact on your average is the same. Therefore, the answer is that it doesn’t matter; you should go to the pub after as many games as possible, regardless of the result.
But the question is phrased as: “Is it better to go to the pub after losses, draws or wins and why?” implying that there is a strategic choice to be made. But according to the math, there’s no difference. However, this seems counterintuitive. Maybe there’s a different way to model it.
Wait, perhaps the confusion is that the 0.5 points for going to the pub are awarded per game, not per pub visit. So each game you go to the pub, you get an extra 0.5 points. Therefore, if you go to the pub after a win, you get 3 + 0.5 = 3.5 for that game. If you go after a loss, you get 0 + 0.5 = 0.5. Therefore, each pub visit gives you a different amount based on the game result. But since the 0.5 is added to the base points, the total points depend on where you add them.
Wait, but if you have to choose which games to add the 0.5 to, then adding them to higher base point games would give a higher total. For example, adding 0.5 to a win gives you 3.5, whereas adding 0.5 to a loss gives 0.5. Therefore, the total points would be higher if you add the 0.5 to higher base games. Hence, to maximize total points (and thus average), you should allocate your pub visits to games where you have more base points, i.e., wins, then draws, then losses.
Yes, that makes sense. If you have a limited number of pub visits, each giving +0.5 points to that game’s total, then to maximize your overall points, you should allocate those +0.5 to the games where you already have the highest base points. Because adding 0.5 to a higher base game increases your total more than adding it to a lower base game. Wait, but 0.5 is 0.5 regardless. But no, the total points are additive: each pub visit adds 0.5. Therefore, regardless of which game you add it to, the total points increase by 0.5. Therefore, the total points are the same, hence the average is the same.
But this contradicts the previous reasoning. Wait, perhaps I need to think in terms of optimizing given a limited number of pub visits. Let’s formalize it.
Let’s suppose you have W wins, D draws, and L losses. Total games N = W + D + L. Total base points = 3W + D + 0L = 3W + D.
If you go to the pub after K games, you get an additional 0.5*K points. Total points = 3W + D + 0.5K. Average = (3W + D + 0.5K)/N.
Therefore, to maximize the average, you need to maximize K, the number of pub visits. However, if K is fixed (i.e., you can only go to the pub a certain number of times regardless of game outcomes), then the average is determined by K. But if K is not fixed, and you can choose K, but the only constraint is that you cannot go to all games (K < N), then you should maximize K (go to as many pubs as possible) to maximize the average.
But the question is: given that you cannot go to the pub every game, should you go after losses, draws, or wins? If K is fixed (i.e., you have to miss a certain number of games), then the allocation of pub visits does not affect total points, as each pub visit gives the same 0.5. Therefore, the answer would be it doesn’t matter.
But this seems to ignore that the base points vary per game. Wait, but total points are 3W + D + 0.5K. So regardless of which games you choose to add K, it’s still 0.5K. Therefore, the allocation doesn’t matter. Therefore, the answer is that it doesn’t matter; you can go to the pub after any games, and your average will be the same.
But wait, this can’t be right. For example, suppose you have two games: one win and one loss. You can go to the pub once. If you go after the win, total points are 3.5 + 0 = 3.5, average 1.75. If you go after the loss, total points are 3 + 0.5 = 3.5, average 1.75. Same result. Therefore, it doesn’t matter. Similarly, for two games: two wins. If you go to the pub once, total points 3.5 + 3 = 6.5, average 3.25. If you go to the pub once, same result. Wait, no, if you have two wins and go to the pub once, total points are 3.5 + 3 = 6.5. If you could go to the pub after both, it would be 7. So the number of pub visits matters, but the allocation doesn’t.
Therefore, the answer is that it doesn’t matter which games you go to the pub after; the average depends only on the number of pub visits. Therefore, since you can’t go to all, you should go to as many as possible, regardless of the game outcome.
But the question is specifically asking whether it’s better to go after losses, draws, or wins. Given that the allocation doesn’t matter, the answer would be that it doesn’t matter, but the options given are losses, draws, or wins. However, maybe the question assumes that you have to skip some pub visits, and you should skip the ones that give you the least marginal benefit.
Alternatively, maybe there’s a different angle. Consider that the points per game is your total points divided by games played. If you go to the pub, you get an extra 0.5 points, but you still played the same number of games. Therefore, the 0.5 is just added to the total. Therefore, the average increases by 0.5/N for each pub visit. Therefore, to maximize the average, you need to maximize the number of pub visits. Therefore, regardless of when you go, each pub visit adds 0.5/N to your average. Therefore, it’s optimal to go to the pub as much as possible. If you can’t go to all, then it doesn’t matter which ones you skip, because each skipped pub visit costs you 0.5/N in average. Therefore, to minimize the loss, you should skip the pub visits that you value least. But since all pub visits give the same 0.5 points, there’s no difference.
But maybe the confusion is that the pub visit is after the game, so if you lose, you might not feel like going, but the question is about strategy, not feelings. The mathematical answer is that it doesn’t matter. However, perhaps the question is trying to get at the idea that if you go to the pub after a loss, you’re adding 0.5 to a 0, making it 0.5, which is a higher percentage increase, whereas after a win, it’s 3 to 3.5, which is a smaller percentage increase. But since the league is sorted by points per game, not percentage improvement, the absolute increase is what matters, and 0.5 is 0.5 regardless.
Therefore, the answer is that it doesn’t matter; you should go to the pub as often as possible. However, since the options are losses, draws, or wins, perhaps the intended answer is to go after losses, because adding 0.5 to a loss gives you some points instead of none, whereas adding to a win or draw is just increasing already existing points. But in terms of pure points per game, it doesn’t matter. However, maybe psychologically, getting some points after a loss is better, but mathematically, it’s the same.
Wait, but if you think about it in terms of the impact on your average, adding 0.5 to a game where you got 0 points increases that game’s contribution from 0 to 0.5, which is significant. Whereas adding 0.5 to a win only increases it from 3 to 3.5. However, over the entire average, each game is weighted equally. Therefore, each 0.5 addition affects the average by the same amount. For example, in a season of N games, each pub visit adds 0.5/N to your average. Therefore, whether you add it to a win, loss, or draw, it’s the same.
Therefore, the conclusion is that the allocation doesn’t matter. However, the possible answers are losses, draws, or wins, so perhaps the intended answer is to go after losses because that’s when you get the most “value” from the pub points, turning a 0 into 0.5, whereas for a win it’s just a small increase. But in terms of absolute points per game, it’s still the same. The question is about maximizing points per game, not about making the most of each pub visit. Since points per game is total points divided by games, and total points are increased by 0.5 per pub visit regardless of when you go, there’s no difference.
But maybe in a different sense: if you have a limited number of pub visits, say M, and you want to allocate them to maximize your total points, but your total points are fixed as 3W + D + 0.5M. Therefore, since W and D are fixed (based on your actual results), the total points depend only on M. Therefore, you should maximize M, i.e., go to the pub as much as possible. Therefore, if you have to skip some, it doesn’t matter which ones you skip. Therefore, the answer is that it doesn’t matter.
However, this seems to contradict the intuitive answer that you should allocate the pub visits to the losses to get something from them. But mathematically, it’s the same. The confusion might come from thinking that the 0.5 is multiplied by the result, but it’s not; it’s a flat 0.5.
Therefore, the correct answer is that it doesn’t matter; you should go to the pub as often as possible. But since the question gives the options of losses, draws, or wins, and asks which is better, maybe there’s a different line of reasoning.
Wait, perhaps in the case of a tiebreaker. If two players have the same points per game, maybe the tiebreaker is total points, then whoever has more wins, etc. But the question doesn’t mention tiebreakers, only that the league is sorted by points per game.
Alternatively, maybe the misunderstanding is in the problem statement. If going to the pub gives you half a point for the game, regardless of the result, then whether you go to the pub affects that game’s points. If you don’t go, you get the base points; if you go, you get base points + 0.5. Therefore, each game you go to the pub effectively adds 0.5 to your total. Therefore, to maximize your total, you should go to the pub as often as possible. If you can’t go to all, the ones you skip will miss out on 0.5 points each. Therefore, it doesn’t matter which ones you skip; each skipped game costs you 0.5 points. Therefore, to minimize the loss, go to as many as possible. But since you have to skip some, it’s irrelevant which ones.
But the question is, “If I cannot make it to the pub for every single game. Is it better to go to the pub after losses, draws or wins and why?” The answer is that it doesn’t matter because each pub attendance gives the same fixed bonus. However, the options are losses, draws, or wins, so maybe the intended answer is that you should go after losses to gain the most relative benefit (turning a 0 into 0.5), but in absolute terms, it’s the same as going after a win or draw.
But in terms of points per game, which is an absolute measure, not relative, the 0.5 adds the same. Therefore, the answer is it doesn’t matter. However, this is a math problem, and usually, such questions have a definitive answer. Given that, perhaps the intended answer is to go after losses because the marginal gain is higher (infinite percentage increase), but in absolute terms, it’s the same.
Alternatively, maybe there’s a misconception that points per game is calculated differently, like average per result type. But no, it’s total points divided by total games.
Wait, let’s think of another angle. Suppose you have a higher points per game if you have more high-scoring games. For example, if you have two games:
Game 1: Win with pub (3.5)
Game 2: Loss without pub (0)
Average: 1.75Alternatively:
Game 1: Win without pub (3)
Game 2: Loss with pub (0.5)
Average: 1.75Same average. Therefore, it doesn’t matter. Therefore, the answer is that it doesn’t matter; you should go to as many pubs as possible. But since you have to miss some, it doesn’t matter which ones you miss.
But the problem might be expecting that since a win gives 3 points, adding 0.5 is a smaller percentage increase, while adding 0.5 to a loss is a larger relative increase. However, in terms of absolute points per game, they contribute the same.
Therefore, the answer is that it doesn’t matter; all choices yield the same points per game. However, given the options, maybe the intended answer is to go after losses because that’s when the 0.5 makes the biggest difference, but mathematically, it’s incorrect.
Alternatively, maybe the reasoning is that you should maximize the number of games where you get at least some points. For example, if you go to the pub after losses, you turn a 0 into 0.5, so you have fewer games with 0 points. However, in terms of points per game, it’s the same. But in some leagues, tiebreakers might consider number of wins, etc., but the question doesn’t mention that.
Given the problem as stated, where the league is sorted solely by points per game, and points per game is total points divided by total games, the correct answer is that it does not matter whether you go to the pub after wins, draws, or losses; the effect on your average will be the same. Each pub visit adds 0.5 to your total points, regardless of the result of the game.
Therefore, the advice is to go to the pub as often as possible, irrespective of the game’s outcome. However, since the question asks “Is it better to go to the pub after losses, draws or wins and why?”, the answer must be one of the three. Given that, but knowing that mathematically they are equivalent, the answer might be expected to be either “it doesn’t matter” or based on a misunderstanding. But since the options are losses, draws, or wins, perhaps the answer is losses, because you gain 0.5 instead of 0, which is better than adding to an already higher score. But in reality, it’s the same.
However, strictly mathematically, the answer is that it doesn’t matter. But if forced to choose, perhaps the intended answer is losses.
The key is that each pub visit adds a fixed 0.5 points to your total, regardless of the game’s result. Since points per game is calculated as total points divided by the number of games played, it doesn’t matter whether you go to the pub after wins, draws, or losses. Each pub visit contributes the same amount (0.5 points) to your total, so the order of priority (losses, draws, or wins) has no impact on your points-per-game average.
However, if forced to choose, one could argue that going after losses is marginally more beneficial psychologically or in terms of “minimizing damage” (turning 0 points into 0.5), but mathematically, all choices are equivalent.
Answer: It does not matter. You should go to the pub as often as possible, since each visit adds 0.5 points regardless of the game’s result. The points-per-game average depends only on the number of pub visits, not the outcomes of the games you attend.
Claude Sonnet 3.7 (think mode) — thought a lot about this
Thoughts
In this scenario, I need to analyze what strategy maximizes points per game (PPG) when a player can’t attend the pub after every game but wants to optimize their PPG.
The point system works as follows:
Win: 3 points
Draw: 1 point
Loss: 0 points
Going to pub: 0.5 points (additional)
The PPG is calculated as total points divided by games played.Let’s consider the different scenarios of when to attend the pub:
Scenario 1: Go to pub after wins
For wins with pub attendance: 3 + 0.5 = 3.5 points
For draws without pub attendance: 1 point
For losses without pub attendance: 0 pointsScenario 2: Go to pub after draws
For wins without pub attendance: 3 points
For draws with pub attendance: 1 + 0.5 = 1.5 points
For losses without pub attendance: 0 pointsScenario 3: Go to pub after losses
For wins without pub attendance: 3 points
For draws without pub attendance: 1 point
For losses with pub attendance: 0 + 0.5 = 0.5 pointsTo determine the optimal strategy, we need to see which scenario yields the highest points per game. This will depend on the frequency of wins, draws, and losses.
Let’s say:
w = number of wins
d = number of draws
l = number of losses
Total games = w + d + l
For scenario 1 (pub after wins):
Total points = 3.5w + d
PPG = (3.5w + d) / (w + d + l)For scenario 2 (pub after draws):
Total points = 3w + 1.5d
PPG = (3w + 1.5d) / (w + d + l)For scenario 3 (pub after losses):
Total points = 3w + d + 0.5l
PPG = (3w + d + 0.5l) / (w + d + l)Now, let’s compare these scenarios. Since we’re dividing by the same denominator (total games), we can just compare the numerators to see which one gives more total points.
Comparing scenario 1 and 2:
3.5w + d versus 3w + 1.5d
The difference is 0.5w — 0.5dIf w > d, then scenario 1 is better.
If w < d, then scenario 2 is better.
If w = d, they’re equivalent.Comparing scenario 1 and 3:
3.5w + d versus 3w + d + 0.5l
The difference is 0.5w — 0.5lIf w > l, then scenario 1 is better.
If w < l, then scenario 3 is better.
If w = l, they’re equivalent.Comparing scenario 2 and 3:
3w + 1.5d versus 3w + d + 0.5l
The difference is 0.5d — 0.5lIf d > l, then scenario 2 is better.
If d < l, then scenario 3 is better.
If d = l, they’re equivalent.Let’s think about this more generally. In most league games, we would typically expect that w + d + l = constant (total games), and we want to maximize our points.
Actually, I think I can approach this more simply. The question is about maximizing the “bang for your buck” when you choose to go to the pub.
When you go to the pub, you add 0.5 points to your total. The key insight is that these 0.5 points will have the biggest percentage increase on your smallest point totals:
For a win (3 points), adding 0.5 is a 16.7% increase (0.5/3 = 0.167)
For a draw (1 point), adding 0.5 is a 50% increase (0.5/1 = 0.5)
For a loss (0 points), adding 0.5 is an infinite percentage increase (0.5/0 → ∞)
From this perspective, you get the biggest relative boost by going to the pub after losses, then draws, then wins.But there’s another way to look at it: we want to maximize our points per game. Since the total number of games remains the same regardless of our pub attendance strategy, we want to maximize the total points.
Let’s say you can only go to the pub a fixed number of times, let’s call it p, where p is less than the total number of games. Then:
If you go after p wins, your total is: 3w + d + 0.5p
If you go after p draws, your total is: 3w + d + 0.5p
If you go after p losses, your total is: 3w + d + 0.5p
In all cases, the total is the same! This is because we get the same 0.5 points for going to the pub regardless of the game outcome.Actually, I think I need to be more careful. If we have a fixed number of wins, draws, and losses, and we can only go to the pub after a subset of games, then the strategy indeed matters.
Let’s be more precise:
Total games = w + d + l
You can attend the pub p times, where p < w + d + l
We want to maximize total points
For a game where you win and attend the pub: 3 + 0.5 = 3.5 points
For a game where you win but don’t attend the pub: 3 points
Difference: 0.5 pointsFor a game where you draw and attend the pub: 1 + 0.5 = 1.5 points
For a game where you draw but don’t attend the pub: 1 point
Difference: 0.5 pointsFor a game where you lose and attend the pub: 0 + 0.5 = 0.5 points
For a game where you lose but don’t attend the pub: 0 points
Difference: 0.5 pointsSince the difference is the same (0.5 points) regardless of whether you win, draw, or lose, it doesn’t matter which games you choose to attend the pub after. The total points will be the same.
But that doesn’t sound right either, because the question is asking for a strategy…
Oh wait, I think I misunderstood the problem. Let me re-read the question.
“The league is sorted by the highest to lowest points per game. This is your total points divided by number of games played.”
So the ranking is based on PPG, not total points. This is a crucial detail.
“If I cannot make it to the pub for every single game. Is it better to go to the pub after losses, draws or wins and why?”
The question is asking which strategy maximizes PPG when you can’t attend the pub every time.
Let’s reconsider. If you attend the pub after a game, you earn an additional 0.5 points for that game. The points per game (PPG) is:
PPG = (Points from wins + Points from draws + Points from losses + Points from pub attendance) / Total games
Since the total number of games is constant, we want to maximize the numerator. Going to the pub adds 0.5 points to the game’s outcome.
The key insight is that the 0.5 points from the pub is a constant addition to any game’s outcome. Whether you win, draw, or lose, you get the same 0.5 points for going to the pub.
Since we’ve established that going to the pub adds the same 0.5 points regardless of the game’s outcome, it doesn’t matter strategically whether you go after wins, draws, or losses. Your total points will be the same.
But wait, the question asks if “it is better to go to the pub after losses, draws or wins” if you “cannot make it to the pub for every single game.” This implies we need to choose a specific outcome after which to attend.
Let me re-examine this from a different angle:
Let’s say:
P_w = probability of winning
P_d = probability of drawing
P_l = probability of losing
x_w = proportion of wins after which you go to the pub (0 to 1)
x_d = proportion of draws after which you go to the pub (0 to 1)
x_l = proportion of losses after which you go to the pub (0 to 1)
Your constraint is that you can only go to the pub a fraction p of the time, where p < 1.
This means:
P_w * x_w + P_d * x_d + P_l * x_l = pYour expected points per game would be:
PPG = 3P_w + P_d + 0.5(P_wx_w + P_dx_d + P_l*x_l)Since P_wx_w + P_dx_d + P_lx_l = p, this becomes:
PPG = 3P_w + P_d + 0.5*pThe term 0.5*p is constant, so it doesn’t matter how you distribute your pub attendance among wins, draws, and losses.
But that still doesn’t seem right…
Wait, there’s a flaw in my reasoning. I’ve been assuming that the constraint is on the total number of pub visits. But if the constraint is “I cannot make the pub for every single game,” then we’re optimizing which types of games to prioritize for pub attendance.
If you can only attend the pub after certain game outcomes (e.g., only after wins, or only after draws, or only after losses), then the strategy matters.
Let’s reconsider with a more concrete example to clarify.
Imagine you play 10 games:
5 wins
3 draws
2 losses
Strategy 1: Go to pub after all wins5 wins with pub: 5 * (3 + 0.5) = 17.5 points
3 draws without pub: 3 * 1 = 3 points
2 losses without pub: 2 * 0 = 0 points
Total: 20.5 points
PPG: 20.5 / 10 = 2.05
Strategy 2: Go to pub after all draws5 wins without pub: 5 * 3 = 15 points
3 draws with pub: 3 * (1 + 0.5) = 4.5 points
2 losses without pub: 2 * 0 = 0 points
Total: 19.5 points
PPG: 19.5 / 10 = 1.95
Strategy 3: Go to pub after all losses5 wins without pub: 5 * 3 = 15 points
3 draws without pub: 3 * 1 = 3 points
2 losses with pub: 2 * (0 + 0.5) = 1 point
Total: 19 points
PPG: 19 / 10 = 1.9
In this example, Strategy 1 (pub after wins) gives the highest PPG.But wait, I made an error. Let’s make sure I understand the constraint correctly.
If the constraint is that you have a limited number of pub visits, and you want to know which game outcomes to prioritize for those visits, then there are two ways to interpret this:
You can only attend the pub after specific outcomes (e.g., only after wins).
You have a fixed number of pub visits and can choose which game outcomes to use them on.
Let’s interpret it as the second option: you have a fixed budget of pub visits and want to maximize your PPG.Given a fixed number of pub visits, it doesn’t matter which game outcomes you use them on. The additional 0.5 points is the same whether you win, draw, or lose. So your total points, and consequently your PPG, will be the same.
But that doesn’t align with the question’s wording. The question asks if it’s better to go to the pub after losses, draws, or wins. This suggests we’re choosing one specific outcome to prioritize.
I guess we need to interpret the constraint as: you can only attend the pub after one type of game outcome (win, draw, or loss).
Under this interpretation, and assuming you have an equal probability of each outcome, the expected PPG would be:
Pub after win: P_w * (3 + 0.5) + P_d * 1 + P_l * 0
Pub after draw: P_w * 3 + P_d * (1 + 0.5) + P_l * 0
Pub after loss: P_w * 3 + P_d * 1 + P_l * (0 + 0.5)
If we assume equal probabilities for simplicity (1/3 each):Pub after win: 1/3 * 3.5 + 1/3 * 1 + 1/3 * 0 = 1.5
Pub after draw: 1/3 * 3 + 1/3 * 1.5 + 1/3 * 0 = 1.5
Pub after loss: 1/3 * 3 + 1/3 * 1 + 1/3 * 0.5 = 1.5
They all give the same expected PPG under these assumptions.But the real question is: which strategy maximizes PPG in a realistic setting where the probabilities are not equal?
The expected points for the different strategies are:
Pub after win: 3P_w + 0.5P_w + P_d + 0P_l = 3.5P_w + P_d
Pub after draw: 3P_w + P_d + 0.5P_d + 0P_l = 3P_w + 1.5*P_d
Pub after loss: 3P_w + P_d + 0P_l + 0.5P_l = 3P_w + P_d + 0.5*P_l
Now we can compare these based on realistic probabilities.I think I’ve been overthinking this. Let’s step back.
The question is asking which game outcome (win, draw, or loss) you should prioritize for pub attendance if you can’t attend every time.
Let’s look at the point increase for each outcome:
Win: 3 points without pub, 3.5 points with pub. Increase: 0.5 points.
Draw: 1 point without pub, 1.5 points with pub. Increase: 0.5 points.
Loss: 0 points without pub, 0.5 points with pub. Increase: 0.5 points.
The absolute increase is the same for each outcome: 0.5 points. But the relative increase is different:For a win, the relative increase is 0.5/3 = 16.7%.
For a draw, the relative increase is 0.5/1 = 50%.
For a loss, the relative increase is 0.5/0 (undefined, but effectively infinite).
From a relative perspective, going to the pub after a loss gives the biggest percentage increase in points.But does this translate to a higher PPG? Let’s see with an example.
Imagine you play 10 games:
5 wins
3 draws
2 losses
And you can only attend the pub after 5 games total.Strategy 1: Prioritize pub after losses, then draws, then wins
2 losses with pub: 2 * (0 + 0.5) = 1 point
3 draws with pub: 3 * (1 + 0.5) = 4.5 points
0 wins with pub: 0
5 wins without pub: 5 * 3 = 15 points
Total: 20.5 points
PPG: 20.5 / 10 = 2.05
Strategy 2: Prioritize pub after wins, then draws, then losses5 wins with pub: 5 * (3 + 0.5) = 17.5 points
0 draws with pub: 0
0 losses with pub: 0
3 draws without pub: 3 * 1 = 3 points
2 losses without pub: 2 * 0 = 0 points
Total: 20.5 points
PPG: 20.5 / 10 = 2.05
Strategy 3: Prioritize pub after draws, then wins, then losses3 draws with pub: 3 * (1 + 0.5) = 4.5 points
2 wins with pub: 2 * (3 + 0.5) = 7 points
0 losses with pub: 0
3 wins without pub: 3 * 3 = 9 points
2 losses without pub: 2 * 0 = 0 points
Total: 20.5 points
PPG: 20.5 / 10 = 2.05
All strategies give the same PPG. This is because the pub attendance adds a flat 0.5 points regardless of the game’s outcome.Wait, I think I’ve been misunderstanding the problem. Let’s re-read it:
“Individually we each get 3 pts for a win. 1 pt for a draw. 0 pts for a loss. We also receive half a point if we go to the pub after the game.”
The key phrase is “if we go to the pub after the game.” This means you have the choice after each game to go to the pub or not. It’s not about committing to a strategy in advance like “I’ll always go to the pub after losses.”
The constraint “If I cannot make it to the pub for every single game” suggests that for some games, you won’t be able to attend the pub (e.g., due to other commitments).
If you have to decide on a strategy for when you can make it to the pub, and the goal is to maximize PPG, then the question is valid: Should you prioritize going to the pub after wins, draws, or losses?
With this clarification, let’s consider a scenario where you can attend the pub a limited number of times. Since the pub attendance adds a flat 0.5 points regardless of the game’s outcome, your total points will be the same no matter which games you choose for pub attendance.
Therefore, from a pure points perspective, it doesn’t matter whether you go to the pub after wins, draws, or losses. Your PPG will be the same.
However, there’s a psychological aspect to consider. If you’ve just lost a game, going to the pub might help you decompress and maintain team spirit, which could indirectly improve your performance in future games. Similarly, celebrating wins at the pub can boost morale. These psychological factors might indirectly affect your long-term PPG, but they’re outside the scope of the mathematical analysis.
Actually, I think we need to reconsider the constraint more carefully.
The question says “If I cannot make it to the pub for every single game.” This suggests there are some specific game days when you can’t attend the pub, not that you’re choosing one type of outcome to prioritize.
Under this interpretation, the choice is not about which outcome to prioritize for pub attendance, but about which attendance strategy is best if you’re constrained by external factors.
I think my initial interpretation was correct. If you can only attend the pub after, say, 50% of your games, should you prioritize going after wins, draws, or losses to maximize your PPG?
Let’s revisit the scenario with 10 games:
5 wins
3 draws
2 losses
If you can only attend the pub after 5 games, here are the different strategies:Strategy 1: Prioritize pub after losses, then draws, then wins
Total points: 53 (wins without pub) + 01 (draws without pub) + 00 (losses without pub) + 03.5 (wins with pub) + 31.5 (draws with pub) + 20.5 (losses with pub) = 15 + 4.5 + 1 = 20.5
PPG: 20.5 / 10 = 2.05Strategy 2: Prioritize pub after wins, then draws, then losses
Total points: 03 (wins without pub) + 31 (draws without pub) + 20 (losses without pub) + 53.5 (wins with pub) + 01.5 (draws with pub) + 00.5 (losses with pub) = 17.5 + 3 = 20.5
PPG: 20.5 / 10 = 2.05Strategy 3: Prioritize pub after draws, then wins, then losses
Total points: 23 (wins without pub) + 01 (draws without pub) + 20 (losses without pub) + 33.5 (wins with pub) + 31.5 (draws with pub) + 00.5 (losses with pub) = 6 + 10.5 + 4.5 = 21
PPG: 21 / 10 = 2.1Wait, that can’t be right. Let me recalculate.
Strategy 1: Prioritize pub after losses, then draws, then wins
2 losses with pub: 2 * 0.5 = 1 point
3 draws with pub: 3 * 1.5 = 4.5 points
0 wins with pub: 0
5 wins without pub: 5 * 3 = 15 points
0 draws without pub: 0
0 losses without pub: 0
Total: 1 + 4.5 + 15 = 20.5 points
PPG: 20.5 / 10 = 2.05
Strategy 2: Prioritize pub after wins, then draws, then losses5 wins with pub: 5 * 3.5 = 17.5 points
0 draws with pub: 0
0 losses with pub: 0
0 wins without pub: 0
3 draws without pub: 3 * 1 = 3 points
2 losses without pub: 2 * 0 = 0 points
Total: 17.5 + 3 = 20.5 points
PPG: 20.5 / 10 = 2.05
Strategy 3: Prioritize pub after draws, then wins, then losses0 losses with pub: 0
3 draws with pub: 3 * 1.5 = 4.5 points
2 wins with pub: 2 * 3.5 = 7 points
3 wins without pub: 3 * 3 = 9 points
0 draws without pub: 0
2 losses without pub: 2 * 0 = 0 points
Total: 4.5 + 7 + 9 = 20.5 points
PPG: 20.5 / 10 = 2.05
All strategies result in the same PPG of 2.05. This confirms that, from a pure points perspective, it doesn’t matter whether you prioritize going to the pub after wins, draws, or losses.The reason is that going to the pub adds a flat 0.5 points to your total, regardless of whether you win, draw, or lose. Since you’re adding the same 0.5 points to different outcomes, the total points remain the same.
To illustrate, let’s simplify the example to just two games: one win and one loss. If you can only go to the pub after one game, here are the two strategies:
Strategy A: Pub after win
1 win with pub: 3 + 0.5 = 3.5 points
1 loss without pub: 0 points
Total: 3.5 points
PPG: 3.5 / 2 = 1.75
Strategy B: Pub after loss1 win without pub: 3 points
1 loss with pub: 0 + 0.5 = 0.5 points
Total: 3.5 points
PPG: 3.5 / 2 = 1.75
Both strategies yield the same PPG because the pub attendance adds the same 0.5 points regardless of the game’s outcome.So, it doesn’t matter whether you go to the pub after wins, draws, or losses from a mathematical standpoint. The PPG will be the same.
Actually, I realize there’s one aspect I haven’t considered: the distribution of games.
In reality, you won’t have an exactly even distribution of wins, draws, and losses. There will be variation. If you commit to a strategy like “always go to the pub after losses,” your PPG could vary based on how many losses you actually incur.
For instance, if you commit to “pub after losses” but end up with no losses, you won’t get any pub bonuses. Whereas if you commit to “pub after wins” and end up with many wins, you’ll get more pub bonuses.
But this is a different question: Which strategy is the most robust against uncertainty in game outcomes? That’s a more complex question that depends on the expected distribution of wins, draws, and losses, and the variability of these outcomes.
If we’re just answering the original question, then mathematically, it doesn’t matter which strategy you choose. Your PPG will be the same.
But actually, wait, I haven’t considered one aspect properly. I think I’ve been over-complicating this.
Let’s re-interpret the question:
“If I cannot make it to the pub for every single game. Is it better to go to the pub after losses, draws or wins and why?”
I believe the question is asking not which outcome to prioritize but specific strategies like “always go after losses,” “always go after draws,” or “always go after wins.”
For instance, the “go after losses” strategy means you only go to the pub when you lose. If you don’t lose, you don’t go to the pub.
The “go after wins” strategy means you only go to the pub when you win. If you don’t win, you don’t go to the pub.
Similarly, the “go after draws” strategy means you only go to the pub when you draw.
Under these interpretations, the PPG for each strategy would be:
Strategy 1: Go to pub after losses only
Expected PPG = 3P_w + P_d + 0.5P_lStrategy 2: Go to pub after draws only
Expected PPG = 3P_w + P_d + 0.5P_dStrategy 3: Go to pub after wins only
Expected PPG = 3P_w + 0.5P_w + P_dNow, which strategy is better depends on the probabilities P_w, P_d, and P_l.
If P_w is high, then Strategy 3 might give more total pub bonuses. If P_l is high, then Strategy 1 might give more total pub bonuses. Similarly, if P_d is high, then Strategy 2 might be better.
But the question is not about which strategy gives more total pub bonuses, but which gives the highest PPG.
Let’s compare the expected PPG for the different strategies:
Strategy 1 (pub after losses): 3P_w + P_d + 0.5P_l
Strategy 2 (pub after draws): 3P_w + P_d + 0.5P_d = 3P_w + 1.5P_d
Strategy 3 (pub after wins): 3P_w + 0.5P_w + P_d = 3.5*P_w + P_dTo determine which is the highest, we need to compare them:
Strategy 1 vs Strategy 2:
3P_w + P_d + 0.5P_l versus 3P_w + 1.5P_d
The difference is 0.5P_l — 0.5P_dIf P_l > P_d, then Strategy 1 is better.
If P_l < P_d, then Strategy 2 is better.
If P_l = P_d, they’re equivalent.Strategy 1 vs Strategy 3:
3P_w + P_d + 0.5P_l versus 3.5P_w + P_d
The difference is 0.5P_l — 0.5*P_wIf P_l > P_w, then Strategy 1 is better.
If P_l < P_w, then Strategy 3 is better.
If P_l = P_w, they’re equivalent.Strategy 2 vs Strategy 3:
3P_w + 1.5P_d versus 3.5P_w + P_d
The difference is 0.5P_d — 0.5*P_wIf P_d > P_w, then Strategy 2 is better.
If P_d < P_w, then Strategy 3 is better.
If P_d = P_w, they’re equivalent.In most sports or games, typically P_w > P_d and P_w > P_l (i.e., you win more often than you draw or lose). If that’s the case, then Strategy 3 (go to pub after wins) would yield the highest PPG.
But if for some reason you draw more often than you win (P_d > P_w), then Strategy 2 (go to pub after draws) would be better.
And if you lose more often than you win (P_l > P_w), then Strategy 1 (go to pub after losses) would be best.
I think this is the correct interpretation and solution for the question.
If I cannot make it to the pub for every single game, the best strategy (pub after wins, draws, or losses) depends on the frequency of each outcome. If I win more often than I draw or lose, going to the pub after wins gives the highest PPG. If I draw more often than I win or lose, going after draws is best. And if I lose more often than I win or draw, going after losses is optimal.
Actually, I just realized a flaw in my last interpretation. The total number of points with a certain strategy isn’t correct.
Let’s think about it differently.
If I choose the strategy “go to the pub after losses only,” that doesn’t mean I get the pub bonus for every loss. It means I only get the bonus for the losses I can attend.
If the constraint is “I cannot make it to the pub for every single game,” that means there’s some number of games for which I can’t attend the pub, let’s say n_no_pub. The question is, considering this constraint, which games should I prioritize for pub attendance?
If I have n_no_pub games where I can’t attend the pub, then I will attend the pub for the remainder of my games. If I play a total of n_total games, I’ll attend the pub for n_pub = n_total — n_no_pub games.
If I commit to the strategy “go to pub after loss,” I will go to the pub after every loss I can, up to n_pub. If I have more losses than n_pub, I’ll only attend for n_pub of them. If I have fewer losses than n_pub, I’ll attend for all losses and then start attending for other outcomes (draws, then wins) to meet my attendance quota.
Now, the question is asking which strategy maximizes PPG. The expected point increase from going to the pub is simply 0.5 * the number of pub attendances, which is 0.5 * n_pub. This is the same for all strategies!
So, from a purely mathematical standpoint, it makes no difference whether you choose to go to the pub after losses, draws, or wins. Your PPG will be the same.
However, there’s a psychological or behavioral aspect to consider. Going to the pub after a loss might help you cope with the loss and maintain team spirit. Going after a win can be celebratory and reinforce positive feelings. These psychological factors might indirectly affect your long-term performance and PPG, but they’re beyond the scope of the mathematical analysis.
But, I realize there’s another way to approach this, which might make more sense.
If the constraint is something like “I can attend the pub only after my home games,” or “I can only attend the pub on weekdays,” then I have no choice in which game outcomes coincide with my pub availability (assuming no foreknowledge of results).
In this more realistic scenario, let’s check if one of the three possible approaches would be better for my PPG. If I end up with, say, 5 games where I go to the pub and 5 where I don’t, and the pub attendance is random with respect to the game outcome, then would it affect my PPG if those pub attendances came after more of my wins, draws, or losses?
Let’s say out of the 10 games, I have:
5 wins
3 draws
2 losses
And I can only attend the pub after 5 games, which are randomly distributed among the wins, draws, and losses.Would my PPG be higher if those 5 pub attendances coincided with more of my wins, draws, or losses?
If all 5 pub attendances are after wins, I get:
5 wins with pub: 5 * 3.5 = 17.5 points
0 draws with pub: 0
0 losses with pub: 0
0 wins without pub: 0
3 draws without pub: 3 * 1 = 3 points
2 losses without pub: 2 * 0 = 0 points
Total: 17.5 + 3 = 20.5 points
PPG: 20.5 / 10 = 2.05
If all 5 pub attendances are after draws (which is impossible in this example since I only have 3 draws, but for illustrative purposes):0 wins with pub: 0
3 draws with pub: 3 * 1.5 = 4.5 points
2 losses with pub: 2 * 0.5 = 1 point (to reach 5 pub attendances)
5 wins without pub: 5 * 3 = 15 points
0 draws without pub: 0
0 losses without pub: 0
Total: 4.5 + 1 + 15 = 20.5 points
PPG: 20.5 / 10 = 2.05
If all 5 pub attendances are after losses (which is also impossible in this example since I only have 2 losses, but for illustrative purposes):0 wins with pub: 0
3 draws with pub: 3 * 1.5 = 4.5 points (to reach 5 pub attendances after counting 2 losses)
2 losses with pub: 2 * 0.5 = 1 point
5 wins without pub: 5 * 3 = 15 points
0 draws without pub: 0
0 losses without pub: 0
Total: 4.5 + 1 + 15 = 20.5 points
PPG: 20.5 / 10 = 2.05
All scenarios give the same PPG, which again confirms that, from a pure points perspective, it doesn’t matter whether pub attendances coincide more with wins, draws, or losses.But that doesn’t address the real question.
Let’s reconsider. The question is asking about a strategy. If I can’t make it to the pub after every game, should I prioritize going after wins, draws, or losses to maximize my PPG?
The key here is to understand that the question is asking about a prioritization strategy.
Given that going to the pub adds a flat 0.5 points regardless of the game’s outcome, the prioritization strategy won’t affect the PPG. You’ll get the same 0.5 points for each pub attendance, whether it’s after a win, draw, or loss. Even though the relative increase is different (16.7% for a win, 50% for a draw, and infinity% for a loss), the absolute increase is the same: 0.5 points.
So, from a purely mathematical standpoint, it doesn’t matter which strategy you choose. Your PPG will be the same.
But there’s a behavioral or psychological angle that might influence the answer. If you’ve committed to a strategy like “go to the pub after losses,” you might inadvertently change your gameplay to engineer more losses, which would be detrimental to your PPG. Conversely, if you’ve committed to “go to the pub after wins,” it might give you an additional incentive to win, potentially improving your performance. These behavioral effects are beyond the scope of the mathematical analysis but are worth considering.
Given the mathematical indifference and the potential behavioral effects, I’d lean towards recommending “go to the pub after wins” as the strategy.
Actually, I need to be very careful with this. I think I’ve overcomplicated it. The question is simpler than I thought. Let me address it one more time very carefully.
The question asked: “If I cannot make it to the pub for every single game. Is it better to go to the pub after losses, draws or wins and why?”
The key to scoring points is:
3 points for a win
1 point for a draw
0 points for a loss
An additional 0.5 points for going to the pub after any game
The aim is to maximize my PPG, defined as (Total Points) / (Number of Games).If I cannot make it to the pub after every game, I need to decide which games to prioritize for pub attendance.
From a pure points perspective, going to the pub adds a flat 0.5 points to my total, regardless of whether I’ve won, drawn, or lost. Since the increase is the same for any outcome, it doesn’t affect my decision.
But the question asked for the “best” strategy, and this might incorporate factors beyond pure PPG.
If I think about the relative percentage increase in points:
For a win (3 points), the pub adds 0.5 points, which is a 16.7% increase.
For a draw (1 point), the pub adds 0.5 points, which is a 50% increase.
For a loss (0 points), the pub adds 0.5 points, which is an infinite percentage increase.
From this perspective, going to the pub after a loss gives the highest percentage increase in points for that specific game.But the question isn’t asking about percentage increases for specific games. It’s asking which strategy maximizes PPG overall.
Let’s think specifically in terms of expected PPG.
Suppose I have a probability P_w of winning, P_d of drawing, and P_l of losing. And suppose I can attend the pub with a probability P_pub.
If I decide to go to the pub after all games, my expected PPG would be:
E[PPG with pub after all] = P_w * (3 + 0.5) + P_d * (1 + 0.5) + P_l * (0 + 0.5) = P_w * 3.5 + P_d * 1.5 + P_l * 0.5 = 3*P_w + P_d + 0.5But I cannot attend every game, so I need to decide which games to prioritize for pub attendance. There are three strategies:
Strategy 1: Prioritize pub after losses
Probability of going to the pub and losing: min(P_l, P_pub)
Probability of going to the pub and not losing: max(0, P_pub — P_l)Expected points from this strategy:
E[PPG with Strategy 1] = 3P_w + P_d + 0.5min(P_l, P_pub) + 0.5max(0, P_pub — P_l)
= 3P_w + P_d + 0.5*P_pubStrategy 2: Prioritize pub after draws
Probability of going to the pub and drawing: min(P_d, P_pub)
Probability of going to the pub and not drawing: max(0, P_pub — P_d)Expected points from this strategy:
E[PPG with Strategy 2] = 3P_w + P_d + 0.5min(P_d, P_pub) + 0.5max(0, P_pub — P_d)
= 3P_w + P_d + 0.5*P_pubStrategy 3: Prioritize pub after wins
Probability of going to the pub and winning: min(P_w, P_pub)
Probability of going to the pub and not winning: max(0, P_pub — P_w)Expected points from this strategy:
E[PPG with Strategy 3] = 3P_w + P_d + 0.5min(P_w, P_pub) + 0.5max(0, P_pub — P_w)
= 3P_w + P_d + 0.5*P_pubAll three strategies give the same expected PPG: 3P_w + P_d + 0.5P_pub. This confirms that, from a purely mathematical standpoint, it doesn’t matter which strategy you choose. Your expected PPG will be the same.
The reason is simple: going to the pub adds a flat 0.5 points to your total, regardless of whether you win, draw, or lose. Since you’re adding the same 0.5 points to different outcomes, and the constraint is the total number of pub attendances (representated by P_pub), the expected PPG remains the same across strategies.
So, the real answer is, if your goal is to maximize PPG and we will be constrained by external factors when we can attend the pub (e.g., work, family, non-game events), it doesn’t matter which outcome you prioritize for pub attendance. All strategies will lead to the same expected PPG.
My answer to the original question is that, from a mathematical standpoint, it doesn’t matter whether you go to the pub after wins, draws, or losses if you aim to maximize your PPG. The only thing that matters is the number of times you go to the pub. The more you go, the higher your PPG, and that is true regardless of which specific game outcomes you associate with the pub visits.
That said, from a behavioral or psychological perspective, you might want to choose the strategy that aligns with your personal preferences. For instance, if you enjoy celebrating wins or if pub attendance might provide an additional incentive to win, going after wins could be subjectively “better” for you.
Analysis of Points Per Game (PPG) Strategy
Looking at the scoring system:
Win: 3 points
Draw: 1 point
Loss: 0 points
Pub attendance: +0.5 points
Since your league ranks players by points per game (total points divided by games played), the question becomes: which games should you prioritize for pub attendance if you can’t go after every match?
From a purely mathematical perspective, it doesn’t matter whether you go to the pub after wins, draws, or losses. Here’s why:
The pub attendance always adds exactly 0.5 points to your total, regardless of the match outcome. Since the addition is constant, and your total number of games is fixed, your PPG will increase by the same amount regardless of which games you choose for pub visits.
Let’s demonstrate with a simple example of 10 games (5 wins, 3 draws, 2 losses) where you can only attend the pub 5 times:
Strategy 1: Prioritize pub after wins
5 wins with pub: 5 × (3 + 0.5) = 17.5 points
3 draws without pub: 3 × 1 = 3 points
2 losses without pub: 0 points
Total: 20.5 points / 10 games = 2.05 PPG
Strategy 2: Prioritize pub after draws and losses
5 wins without pub: 5 × 3 = 15 points
3 draws with pub: 3 × (1 + 0.5) = 4.5 points
2 losses with pub: 2 × 0.5 = 1 point
Total: 20.5 points / 10 games = 2.05 PPG
However, there is an interesting psychological perspective: going to the pub after losses gives you the highest relative improvement in points for those specific games:
For a win: 0.5/3 = 16.7% increase
For a draw: 0.5/1 = 50% increase
For a loss: 0.5/0 = infinite% increase
So mathematically, it makes no difference to your PPG which games you choose for pub attendance. The optimal strategy is simply to go to the pub as often as possible, regardless of match outcome.
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