A few recent examples:
i. I played Citadels with my little brother this Christmas. I spotted two obvious instances of poor modelling which happened during the game.
The game is complex and I won’t go over all the rules here – it should be noted that the game complexity is probably part of why these errors to be described below were made in the first place. But anyway, we were in a situation where my brother had picked a specific card. Having picked that specific card he had to try to guess which card I had on my hand – if he guessed correctly, I’d lose my turn and the income that turn would generate (which would benefit him and harm me, making him more likely to win the game). There were two obvious candidates; one card generating a potential income of 2 and another other card generating a potential income of 5. He knew I’d taken one of these cards but not which of them I’d picked – if I randomized my draw completely there’d thus be a 50% chance for him to pick the right card. The situation took place during one ’round’ (subgame) of the game, and both of us knew that this would not be the last round in the game. But we did not know how many more rounds were to be played – a conservative estimate would be at least 4 or 5. Whether it would make sense to consider the round to be one round of several in a semi-‘pure’ repeated game or not, and which type of repeated game we’re talking about, depended to some extent on which cards would be picked in future rounds (as I mentioned, the game is complicated – the fundamentals of the stage game can change during gameplay, e.g. I might end up in my brother’s position, i.e. as the player who should guess which card the other player had taken, in a future round); but it would make little sense to consider it a single-shot game.
Now the first thing to note here is that if you consider it a repeated game, it probably doesn’t make a lot of sense not to at least consider to mix strategies. You could probably make an even stronger argument: Consider that if I play ‘2’ (the card giving me an income of 2) with a probability of 100 % my brother would probably pick up on that relatively fast and pick that card every round, and I’d end up with an income of zero – and if I always played ‘5’, he’d always pick 5. So the second person, the one picking the card to be guessed, has to consider adding some uncertainty to the table or he’s probably going to be in trouble. Now let’s think about how one might best mix strategies in this situation. An important theoretical aspect here is that while it’s certainly a finite game, the lenght of the game is still unknown, or at least uncertain, to the players (they do have some idea how long it’ll take to finish the game). This uncertainty adds complexity, and even though only relatively few rounds of the game is left, the game is still much too complex to be solvable by backwards induction by the players while they play the game even if such a solution might exist. Incidentally in the specific game in question when playing that specific subgame I evaluated the costs of reversing the roles of the players (so that I’d get to be the one guessing, which would be a permissible change to the stage game given a specific subgame strategy constellation) to be too high to implement – but my brother didn’t know that.
The first modelling error here was done by me when I was deciding which card to pick. I did pure randomization when I picked my card – basically I shuffled the cards and picked one of the two cards at random. Basically this was just me being stupid, because this is obviously not the best mixed strategy (it’s only optimal in the case where the expected income derived from the two cards are equal). One way to think about this is that a 50% likelihood of picking either card gives you an expected income of 0,5 x 2 + 0,5 x 5 = 3,5 if your opponent also mixes 50/50 – and foolishly I’d considered only that strategic response to my mixing strategy. The problem is that of course the opponent needn’t mix at all! A mixing strategy on his part is obviously dominated by the pure strategy of always picking ‘5’ – if he always picks ‘5’, I end up with an average income of only 1 (I get an income of 2 every second round). I realized this 5 seconds after I’d picked my card..
This is where we get to the second modelling error. My little brother said after that specific round had been played – where he’d picked 5 and I’d gotten lucky and randomly picked 2 (so the inferior strategy did not cost me anything in this specific case) that ‘of course he’d picked 5, it was the dominant strategy’. I thought that this was obviously true in the specific case of a mixing strategy on my part with 50/50 mixing, but that it would not be an optimal response to other mixing strategies with a low probability of playing ‘5’ (nor would it be an optimal response to the pure strategy of 2). I assumed we’d play at least four more rounds, and in that case it would probably be optimal to go with a mixing strategy with a ~30/70% likelihood or something along those lines (i.e. one ‘5’ and 3 ‘2’s in the rounds to come) – I figured that 5 is 2,5 as much as 2, so I should play ‘2’ 2,5 times as often as ‘5’ in equilibrium; i.e. 2,5 ‘2’s for every 1 ‘5’, meaning I should play ‘2’ in 2,5 out of 3,5 rounds, which would be about 70% of the time. I assumed my little brother would mix as well in the rounds to come when I would no longer obviously mix 50/50 and that he’d reach a similar conclusion – that he should pick 5 more often than 2 to minimize my potential income and end up near the (assumed) long-run equilibrium. After the game my little brother made it clear to me that he had not mixed but had played 5 every time, and he stated that he’d picked that strategy because it was ‘the dominant strategy’ and because it would be his best response to any strategy I could come up with. Which it clearly wasn’t.
ii. I went shopping yesterday. I got to the store and it was full of people. I generally dislike shopping when there’s a lot of people around, and I generally avoid this by strategically shopping at times during the day where I know not very many people go shopping. I have previously arrived to a store, decided it was too full of people and postponed my shopping to a later point in time because of that, but yesterday I decided instead to just get it over with fast. When I came back home I remembered that it’s been mentioned in the papers that a lot of people are sick with influenza in Aarhus, and so I realized that I’d just exposed myself to a huge health risk considering how many people were in the store. If asked about this type of stuff before I left my home, I’d have said that such a risk would be completely unacceptable to me, because I have exams before long and thus it would be very inconvenient for me to get sick at this point. If I’d included that health risk in my model, I would not have gone shopping yesterday.
I will often avoid taking public transportation when it’s possible for me to do so due to similar health-related reasons – diseases are easily transmitted in such environments. People often do not remember to include risks like these in their mental models. That’s poor modelling.
Even (reasonably) simple card games and everyday decisions about stuff like when and where to go grocery shopping can include models too complex for humans to handle well; our cognitive limitations are easy to ignore if we don’t think about them, but they’re there just the same. Social dynamics are usually a lot more complex to model than the stuff in the post. Sometimes it seems almost unbelievable to me that people somehow make all this stuff work – taking all those decisions they do on an average day, interacting with all those other people along the way… Given how complex the world is and how even very simple things like a card game can cause us all kinds of problems when we start thinking about them, I find this pretty amazing to think about.