How much information do you need to make a decision? What makes a decision easy or hard, is often framed in the form of how much information you have. In many cases in business life (and to be honest in life in general) is it can be very difficult to make a satisfactory decision much less an optimal decision because we don’t have the full picture. Therefore, we often attribute poor decision making to incomplete information. So the question is how much information does one need to make a good decision? That is I wanted to explore in order to help people understand the challenge of making a decision in a competitive situation.
A natural platform for exploring the impact of decisions has been games. It is such an ideal platform for understanding decision making that there is even a branch of mathematics called “Game Theory” that tries to quantify how one should play a game based on different conditions. One of those conditions is how much information you have in order to make a decision. Some games, all players have complete information on the situation, the canonical example is chess where you know where all the pieces are and all the possible moves are theoretically available for analysis. Other games, you have incomplete information that you have to weigh in order to make a decision, the most popular example is Texas Hold Em Poker where you have some idea of what has been played, and you know what you have, but you do not know what others have. Other games are completely random and you are in the dark, the canonical version is the child’s game “War” which we’ll get to later.
So is it possible to have a game where both parties have complete information about the state of the game and adjust their play accordingly as more information is revealed? Also let’s add can the game also be influenced based on the prior moves. Lastly, can it be an analog to how real life is where people play moves simultaneously and not in an alternating matter like chess where first mover has a slight advantage. In effect, can you create a game where players are equally matched at the beginning of the game, and have complete information about the state of the world in order to make a complete analysis.
That was the objective that I was trying to achieve, with the meta-objective of capturing does complete information about the possible states of world actually enable you to make better decisions?
To that end, I found my inspiration in a children’s card game that most people have played, “War”. In War, a deck of regular playing cards are randomly dealt between players and each player simultaneously reveals their top card. If a player’s card is the highest, they win the other player’s card. If there is a tie, the players lay down two more cards face down and reveal the third card, with the highest winning all the cards. If a tie occurs, they repeat the process. The goal of the game is to win all the cards.
I created a variant of the game called “Fog.” In Fog, each player is given ten cards of increasing value from 1 to 10. Each player then chooses a card to play from their hand and places it face down. Once both players have made their choices their cards are revealed. The player with the highest card gets the points of lower card. In the event of a tie, neither player scores. The goal of the game is to have the higher score after all cards have been played.
For example, if you play the 7 card and your opponent plays the 1 card, you win 1 point. If your opponent plays the 6 card, you gain 6 points. If your opponent plays the 7 card, you get zero points. If your opponent plays an 8 or higher card they get 7 points.
Fog is an attempt to create a card game without randomness from the deck and perfect information for all players. At any point, you know everything about what your opponent has and they know the same as you. You know the history of the game and you can imagine the future of the game with the information present. The only thing matters is the simultaneous choice you and your opponent make. What will you discover with each move?
What is interesting about this game is that while you may choose to win a hand, you may also actively choose to lose a hand as well. This is because each player is aware of the remaining cards their opponent has. So the game has a degree of resource management as well. You not only need to win a hand, but you need to win big. So going for a definite win in any hand, but to only gain a minimum number of points is not satisfactory. One needs to determine a strategy to reveal the strategy of the opponent.
This point winning makes it possible to win more rounds and lose the game. The goal is to not only win a round, but win as close to the max points as possible. Also, when you lose, lose only a little. The maximal winning score is 44 points when you win the most and lose the least.
The name ‘Fog’ comes from this being a variant of the kids card game ‘War’. As a war progresses, you remove the fog and doubt that is ever present at the beginning. The only question, is what will it cost for that clarity?
You can easily play ‘Fog’ against another person with a regular deck of cards. Simply choose two suits and select the 2–10 cards, and use the Ace for 1. When you play your card, play it face down and when both cards have been played, flip the cards and see who won the round. Lay out the cards so you both can see what’s been played. If you want a longer game, simply add more cards or suits. If you play the 10 card variant, it’s best to use match play which is best two out of three, or three out of five games. Or else keep playing until one player has 50, 100 or some agreed upon number.
I also created a simple AI version of the game that is available both on Android and on iPhone for you to try to see if you can come up with a strategy that works.
Android: “https://play.google.com/store/apps/details?id=com.wubashi.fog_card_game
iPhone/iPad: https://apps.apple.com/us/app/fog-card-game/id6450320525
Some observations and open questions for the more nerdy.
Is there a best strategy for playing ‘Fog’? Or maybe more meaningful, can there be an optimal strategy at all? Does it make sense to be cautious and figure out how your opponent plays? Or is it better to blitz and try to win your opponents cards at the beginning? How do you change your play based on what the opponent does?
I did simulations of two players randomly choosing cards, and the results show that both sides will win equally with ties occuring approximately 5% of the time consistently. I also notice that when playing friends who are more similar in mind set to me, (“birds of feather flock together”) the propensity to tie in more rounds seemed to have occurred. It’s an observation, I wonder if others see the same. It suggests it is hard to think differently.
To answer the question of whether there is a better strategy, it seems best to create a tournament between automated strategies and pit them against each other. To determine if a strategy is better, there should be multiple rounds between strategies, perhaps 11 rounds. A strategy wins if it has a higher win percentage and based on the average point score of the winning rounds. This is reminiscent of Robert Axelrod’s tournament to see if there is a best strategy for the iterated prisoner’s dilemma, with the surprising discovery that a simple “tit-for-tat” strategy was in general “best” across strategies submitted. If there is interest in such a tournament for “Fog”, please comment on this post.
The conclusion is that in most strategy curriculums there are heuristics for outfoxing your opponent, but my experiments with “Fog” suggest equally matched adversaries it is difficult to have sustained advantage. Therefore, I am skeptical of managements that say they will leap frog or catch up to opponents in public announcements. It is hard to be multiple times better than an opponent to catch up by trying harder, you are generally sharing the same resource pool. It’s also why brand is so critical because when established it’s not a strict resource and conveys an irrational advantage.