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Subject: [Theory] Analyzing the Psi Game rss

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Adam Sobolewski
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Solving the Psi-Game

Hey guys! I've been reading this forum a lot and really enjoy it. Wanted to give something back, I'm not very good at playing or building decks but I work as a professional poker player and coach with a big focus on game theory so I figured I could use that knowledge in helping you guys with playing the Psi game. Since Caprice is really good and Jinteki is on the rise being good at the Psi game is a good way to increase your edge so might as well take it.

Brief Intro to Game Theory

In the study of games, a famous type of solution is called a Nash equilibrium solution. Playing according to this equilibrium is often (at least in the poker world) called game theory optimal play, or GTO for short.

What is it? Briefly it answers the following question (I'm sticking to two player games here for simplicity):
Imagine that no matter what strategy (this can include mixed strategies that involve choosing multiple options with certain probabilities) you pick, your opponent knows it and picks the best possible counter strategy. What is the best strategy to pick in your spot? The Nash equilibrium strategy.

An alternative definition which gives you more information about the strategy is the following:
A Nash equilibrium is a pair of strategies (one for you, one for your opponent) for which the following property holds: if both players are playing according to the Nash equilibrium (playing GTO), neither player can change their strategy in a way beneficial to themselves. I'll call this property P so that I don't have to rewrite it.

The guy for whom the equilbrium is named after—John Nash (a wonderful movie was made about him called A Beautiful Mind)--also proved that at least one such strategy pair always exists for basically every game.

Now it's not the only type of solution. If happen you know how your opponent is playing then you can just choose a strategy that makes the most vs that particular strategy—this is called the maximally exploitive strategy. In poker terms trying to take advantage of your knowledge of your opponent's strategy is called playing exploitively—you exploit your opponents deviation from GTO in order to make more profit than playing GTO would guarantee you. Notice that when playing exploitively you aren't playing GTO, which then opens you up to being exploited as well since you are necessarily not playing GTO.

GTO strategy is the most profitable strategy only in two situations:
1. Your opponent is playing GTO, or
2. You have no idea at all how your opponent will be playing.
However, it always guarantees some profit no matter what your opponent does—this is why it is optimal in situation 2.

Anyhow, GTO strategies are very useful in many situations and knowing them can also help in knowing how to exploit your opponent because if you know what GTO play looks like, you can figure out in what way your opponent isn't playing GTO and exploit that tendency.

Approximating GTO in the Psi game.

The Psi game in A:NR is particularly amenable to a GTO analysis because of it's simplicity. Many people have already compared it to Rock Paper Scissors and it shares many of the properties of that game. In case you were not aware, the GTO solution to RPS is to play each option an equal probability (33.33% of the time). Check that that's true by making sure it satisfies P.

In order to make the Psi-game amenable to GTO analysis we need to normalize our payoffs. To do so we need to put a credit value to the corporation losing the Psi game and the runner winning. This is one place where we will need to make approximations.

In my analysis the base example Psi game is that of using Caprice Nisei, however you should be able to use the solutions to other examples. In the example of Caprice, I am using the approximation that a runners win is a corps's loss—any points the runner scores means he is closer to victory and the corp therefore closer to losing. I made this equal in magnitude—so if access is effectively worth 3 credits to the runner, it is worth -3 to the corp. There are some exceptions to this (perhaps the value of accessing R&D is negative because there are only snares and no agendas), but we have to make approximations somewhere. Try other arrangements to see how the solution differs. Another underlying approximation is that credits are worth the same to both players. This too can break down in some cases, but overall is a reasonable approximation in my book (the analysis is doable using a factor to devalue one side's credits, but this is not done here—again feel free to do it on your own and post the solutions). I will also always deal with the case where each payment is possible (so both corp and runner have at least 2 credits).

To solve for the Nash equilibrium it is necessary to enumerate the payoffs of each side. After that since the payoff matrix is made up of two 3x3 matrices (one for runner one for corp) the game is relatively small and could be solved by hand. I am too lazy to go through setting up the equations and instead used a free computational game theory tool—Gambit. Google it. It works very well and is relatively simple. If someone is very interested I can post the file of the Gambit tree I used or a screen shot (if they also explain how to post files as I'm still a newb to this forum ). Some issues creep up to do with multiple equlibria (remember I said there is at least 1 guaranteed, but sometimes there are more), but if you just stick to the “find one equilibrium” feature in Gambit you'll be fine and get the “right” equilbria. I don't want to digress about this further though because it gets technical.
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Adam Sobolewski
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Approximate Solutions to Caprice Nisei Psi Game

So let's get some answers! Here are the GTO solutions to the Psi game for different values of access, together with the resulting expected value of the game to the corp and to runner. Just click on the link to see a google doc. The top section has the results for Caprice, the second section will be discussed later on.

https://docs.google.com/spreadsheets/d/1BSklSlsb-36mbWa_7cKB...

(let me know if there's a better way to get this table into the post...)

Expected value is the game value in credits—this is the amount the corp loses on average and the runner gains on average in equilibrium. The runner loss field is the amount the runner lost compared to a direct access—ie this is how much Caprice cost him. Corp gain is how much they gain compared to an access, so how much Caprice gained the corp. Corp swing is the total swing Caprice causes in credits compared to a direct access. % swing is the corp swing divided by the swing caused by a direct access (twice the value of the runner winning the Psi game) to give us an idea of how much the swing was mitigated by Caprice—this will help us later.

Convince yourself that these are Nash equilibria by trying out property P—pick some other strategy and try to see if you can get a higher game value. For example, take the case where the runner wins 2 on access (and corp loses 2)--the solutions here were simple since they ignored the option to pay 2 (try to figure out why, if you're not sure it should be clear at the end of this analysis). The runner knows the corp is paying 1 75% of the time, can't he benefit by always paying 1? Let's compute the value of paying 1: 75% of the time you win the Psi game resulting in winning 2 but paying one for a net of 1 credit. 25% of the time you lose your 1.
.75*1 - .25*1 = .5
Similarly the value of paying 0 is:
.25*2 - .75*0 = .5
So the runner is indifferent to both options! This means he cannot improve his expectation (and of course playing 2 is silly since it guarantees a loss of 2 since we know the corp never plays it). You can compute for the corp similiarly to find that they too are indifferent between paying 0 and 1 if the runner pays 0 75% of the time and 1 25% of the time and again never want to pay 2 (look at the game value if you're not sure why).

The solutions tells us that in general the corp should always pay 2 more often than they pay 0 unless they are trying to exploit their opponent (or the value of access is very low). Paying 1 is usually used around a 1/3 of the time. Conversely, the runner should pay 0 more often than paying 2 and again pays 1 around 1/3 of the time. For low value of access this effect on payments is quite large. At high value of access the solutions approach the same equilibrium as RPS (this should make sense). In the end Caprice mitigates the effect of the swing due to access by at least ½, likely more.

On top of this, the game values ignore the fact that she must be trashed or else she gets to be used again, and that winning for the corp lets you use here again. This adds to her value dramatically—notice that this also makes the stakes of the Psi game effectively higher since losing also means losing Caprice which is worth credits. So in general this tells us that you should always lean towards over valuing access since we should try to fold in the value of having Caprice stay alive.

Overall, we find that in almost any case where access is not trivial Caprice is paying for herself and then some, and often times creates swings as big as account siphon even only being used once. She really is a very good card
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Adam Sobolewski
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The Future Perfect

Notice that the analysis applies exactly to the case of The Future Perfect (TFP) as well. Simply decide the value of having it scored in credits and then you get the solution as well as the value of TFP. Since I'm not great at A:NR it's hard for me to correctly assign the value in credits—clearly sometimes it is infinite since it wins the game, but when it's not game-ending I imagine it to be worth to the runner somewhere in the interval 10-20 points (and therefore a loss to the corp of 10-20 credits). This would mean that TFP mitigates swings between 15 and 25 credits in the cases where it is accessed—a very strong effect, and much stronger than NAPD or a non-lethal Fetal AI!

Further analyzing TFP, in reality the best way to evaluate it's value is simply by converting back and using agenda points—we know that Caprice lowers the value of access swings by a factor of (almost) 2/3 for decently high values of access, and so this means TFP changes access from being worth 3 points to a bit over 1 point—so again it seems to be a strictly stronger effect than NAPD since 2 points is definitely worth more than 4 credits. The optimal play in the Psi game though will still depend on the conversion of points to credits so this is still important—I hope some of the better players can chime in a bit with their intuition on some sort of conversion factors to consider as a function of board state and deck construction so that we can narrow down the solutions a bit. I would assume that an expert could be able to evaluate this value in game and therefore use it to know the GTO Psi game when it's needed.

Account Siphon

The case of Account Siphon is particularly interesting because an (almost) exact solution is achievable since we know exactly what the result of the runner's win is—all we need to approximate is that runner credits and corp credits are worth the same amount. In the previously linked google doc below the Caprice solutions are the solutions for Caprice vs Account Siphon for all of the different corporation starting amounts (assuming Caprice is rezzed). Notice that starting with less than 7 credits leads to interesting play since we can effectively dump credits into the Psi game to lower the runner's value futher. Here Caprice mitigates swings by at least 10 credits, and more when we can dump!
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Adam Sobolewski
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Comments on Proper Use and Exploitative Strategies

Lastly I have a couple comments on the usefulness of these solutions. The property we found earlier in analyzing Caprice for the game with runner access valued at 2 credits demonstrated a curious property—both runner and crop ended up indifferent to their two sensible options. Indifference is actually pervasive in all the solutions—if one side is using a GTO strategy, the other side is indifferent to each of their options—their value is identical. This feature is helps explain a bit why GTO play is sometimes called “unexploitable”. It is also why often in real game situations it is often more profitable to use exploitative strategies since the GTO strategy isn't profiting from your opponents mistakes—once we play GTO we don't care what our opponent does we do exactly the same no matter what. While tempting to go with the “guaranteed win” in reality we expect our opponent does not play GTO, and so if we know in what way they deviate we can look to exploit that mistake and do even better than our game value. So let's explore this a bit more by looking at exploitative strategies.

Maximally exploitative solutions are usually extremal—we'd just always pick one specific option. For example, if we are playing vs an account siphon as corp with 7+ credits, and we believe our opponent plays a RPS strategy of 1/3, 1/3, 1/3 instead of the GTO solution for our specific Psi game then we know they are playing 0 credits with too low a frequency. We exploit this by always choosing 0 credits ourselves.
Our resulting EV is:
1/3*-5 + 2/3*0= -1.66667, 1 credit better than equilibrium.
Our opponent's EV is:
1/3*10 – 1/3*1 – 1/3*2 = 7/3 = 2.3333 the same as equilibrium.

However if we do this our opponent can exploit us by always choosing 0 as well, and suddenly our EV will be terrible and our opponents great (we always get siphoned and on top of that he didn't have to pay in the Psi game!) so we have to be careful—in this situation the gain from exploiting is 1 credit, the loss is 2.3333 credits (and our opponent gains 7.6667 credits). So there actually is a lot of incentive to just stick with the GTO solutions in many cases. There is also room to use a non maximally exploitative strategy that still exploits our opponent—for example we could pay 0 43.33% of the time, pay 1 33.33% of the time and pay 2 23.333% of the time. This would be more profitable than GTO, less obviously exploitative, and have lower regret if we are wrong in our assessment. However we know that the most we can improve our situation by is 1 credit so in the end we might be pushing a small edge on limited information if we pass up using the GTO strategy.

One last comment, I am ignoring the sequential element of the game. As I mentioned earlier, it definitely affects the values of access in the Caprice game. But there are also sequential elements to just playing multiple Psi-games that can sometimes help us gain EV—there are many world class RPS players despite RPS being “solved”, so it might be worth consulting that literature. However in many ways I think the sequential elements here likely aren't very important—in reality most Psi games are different because the board state is different. It will likely be better to make sure to properly incorporate board state over using sequential tricks. On top of that the Psi games are usually not repeated too often. Still, this is a place to explore game theory more if you find yourself interested.

I hope this helps elucidate a bit the details behind Psi games—I think it's useful to know good strategies to use when playing the Psi game (you might want to bring a 100 sided die with you when you play ), as well as useful to have an idea of the value of a Psi game—for example run the analysis for Cerebral Cast to see how profitable it is!). Let me know if you have questions, I'll try to get to them within a week or so. Cheers!
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Timo Kandolin
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Nice! I'm studying mathematics myself and just did a little presentation on Nash some time ago nice to see people mathing the psi-game out also, do pls post the image. I'm interested! Or just personal message me here on bgg.
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Adam Sobolewski
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TimoNator85 wrote:
Nice! I'm studying mathematics myself and just did a little presentation on Nash some time ago nice to see people mathing the psi-game out also, do pls post the image. I'm interested! Or just personal message me here on bgg.


which image? the Gambit game tree?
 
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Timo Kandolin
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Nevermind. I was too quick and only saw your first post before I replied. The doc is perfect. Thx!
 
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Sasha F
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I find this post wonderful! I'm fairly certain I am not fully comprehending the spreadsheet, but hopefully I've gotten some core understanding out of it .

I usually use a more exploitative strategy when choosing Psi games, but it seems to follow the Nash Equations pretty well on Caprice. I think the factor that muddies the Equations most strongly is that runner and corp credits are almost never equivalent. This is based on the inherent asymmetry of the game and afterwards based on the board game state. I would also be interested if someone introduced a credit devaluation factor. Generally I think that corp credits are at least slightly more valuable than runner credits. This is partially due to the original board state, but more so on the relative permanence of runner economy when compared to corp economy.

I find the Siphon based Nash Equations slightly less useful because if I have used Siphon and hit a Caprice, I have already made a more fundamental mistake than needing to worry about playing GTO.
 
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Carl Frodge
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I remember reading somewhere a while ago that with Rock Paper Scissors, more people choose rock first. And that when a person loses in RPS, they're more likely to play whatever beats what they lost to. So if they lost to rock, they're more likely to then play paper, which gives you the knowledge that playing scissors is the optimal play.

I don't know how true or helpful any of that is, but I thought it was interesting.
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R. Fetterkey
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I've noticed several patterns in how people tend to play the Psi game. Unfortunately, sharing them would empower others against me, so I will not do so. However, I will say that sophisticated analysis methods do not seem to be necessary in order to find these patterns-- merely lots of experience.
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Andrew
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fetterkey wrote:
However, I will say that sophisticated analysis methods do not seem to be necessary in order to find these patterns-- merely lots of experience.


Well, the analysis isn't about finding these patterns, it's about determining what is "optimal" even if your opponent knows exactly how you will play.
 
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g k
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I'll state briefly that there is no single optimal play for the psi game because the payoffs are wildly variable. On game point, the Future Perfect Psi game has huge payoffs for both runner and the corp. But if it's the 4th click and the corp needs all of his credits to score out some other installed agenda, biggest payoff for the corp may be forfeiting that psi game altogether.


Half of the payoffs for the psi game are actually hidden from each player. Does bidding 1 or 2 prevent the corp from rezzing tollbooth, archer, or whatever? Does bidding 1 or 2 prevent the runner from installing femme, opus, or corroder? Identifying your opponent's payoffs is the actual game you play when you play the psy game.

Playing the psi game optimally may actually itself be sub optimal. The payoffs for winning the psi game may be smaller than those of retaining privileged information about exactly how many credits you need, or what cards you're afraid of from your opponent.

IMO, it's generally best to reserve your mental effort for identifying degenerate cases of GTO (caprice vs siphon, 3rd click caprice vs 2 credits). Late in the game, if both players are flush with cash, the single credit you might save from playing the psi game optimally will likely be trivial.
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Ryan Angell
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http://www.huffingtonpost.com/2014/05/03/rock-paper-scissors...

interesting study by chinese scientists about rock/paper/scissors. basically the gist is if your opponent is not rolling a dice for his selections he is more likely to repeat a winning selection in the next game and more likely to shift his selection clockwise if he lost the previous bid ie lose with rock next pick is slightly more likely to be paper.

in my experience people who are not rolling a dice for the netrunner psi game seem to have tendencies. runners like zero and 1. i have rarely seen a runner bid 2 unless they are rolling a dice.

corps like 2 on the future perfect and important caprice bids but they try to save money on stuff like cerebral casts, psychic field and snowflake.



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Christopher MacLeod
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GenericKen wrote:
Half of the payoffs for the psi game are actually hidden from each player. Does bidding 1 or 2 prevent the corp from rezzing tollbooth, archer, or whatever? Does bidding 1 or 2 prevent the runner from installing femme, opus, or corroder? Identifying your opponent's payoffs is the actual game you play when you play the psy game.


Hidden payoffs is I think the key point to consider in a useful analysis. The linked table assumes that both players know the payoff for success, which is not even true in the Siphon case! The Siphon analysis overlooks that the Runner may decline to Siphon after winning the psi game if the Corp has sufficiently low credits, and simply access HQ instead with a hidden payoff.

Hidden payoffs are the norm and not the exception in this game, so I think any analysis neglecting that is problematic. The following have all happened in real games:

- The Runner bids on Caprice without knowing what agenda is behind - if it's Fetal or NAPD they can't steal depending on their bid. The Corporation obviously knows what the agenda is.

- The Runner bids on Caprice such that the Corp will be too poor to advance and score the agenda depending on the Corp bid.

- The Runner bids on Caprice such that the Corp will be too poor to use Trick of Light to fast advance an uninstalled agenda, depending on the Corp bid. The Corp knows if he actually has a fast advanceable agenda or not.

- The Runner bids on Caprice hoping to make the Corp spend and be poor, but the Corp secretly has Subliminal Messaging (Shipment from SanSan would work here too) and can feign being too poor and still score.
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Ray Saltrelli
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I apologize if some one already touched on this. The OP was tl;dr.

I think people are missing the boat on the taxation aspect of the psi game. I play a Nisei Division deck with:

- 3x The Future Perfect
- 3x NAPD Contract
- 3x Fetal AI
- 2x Caprice
- 3x Snowflake
- Economically Taxing Ice (Pup, Rainbow, Komainu, Tollbooth, etc)
- Tons of asset-based drip economy
- 3x Encryption Protocol
- 2x Aggressive Secretary (to make them play with Snowflake)

The psi game is not just rock paper scissors. It's an opportunity to exploit a credit advantage. Say I as the corp have 10 credits--not an exorbitant amount but I'm comfortable--and the runner is struggling to maintain his credit pool due to breaking ICE on centrals and trashing assets. Now say the runner is running on a remote with Caprice and NAPD Contract and he only has 5 credits. Spending 2 credits on the psi game (effectively only 1 because I'm Nisei Division) is a guaranteed denial! Either he doesn't spend 2 and is denied by Caprice, or he pays 2 for Caprice and doesn't have enough to steal NAPD Contact.

My point is that if you build a deck around the psi game, then it's often not as simple as rock paper scissors. Board state has a huge impact on the decision and with an identity like Nisei Division the corp has a huge advantage.

From the runner's perspective, he is often going to spend 0 since he wants me to waste credits to keep him out. However, as Nisei Division, this strategy breaks down as I can then spend 1 credit, deny the runner access, and still maintain my credit pool. This forces the runner to consider spending 1 (or even 2 depending on what he thinks I will do) to continue.
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Captain Frisk
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Thank you for posting this. I've been drafting this very article in my head for months.
 
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Thomas Aldershof II
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I would have to say most of this is just Meh.

Because you have no idea what is in the other players head. Not one shred of clarity or fact from what your opponent is thinking.

Here's an example, he could be like "man I just want this match to end, cause I have to goto the bathroom", and bets WHATEVER hoping you'll do the same.

So I just tried to make my point quick and clear.

Aside: Yeah I necro-ed whatever
 
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Michele Lupo
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I suppose that the rational assumption to make is that both players are trying to win the game, otherwise of course ALL discussion about the game goes right out of the window.
 
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Will H.
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Not sure Nash Equilibrium can be applied to this, since it assumes there is a best move for both players. There is not really a fitness function to measure this. For example, in a game I just played:

1. Whoever scores the next agenda wins.

2. I was trying to steal an agenda, and had to deal with Caprice.

3. I only had 1 credit.

4. The Corp had plenty of money, so they would likely spend 2 credits to ensure I don't get access.

Now, is my best move to spend 0 credits, because I know the Corp is going to spend 2?

Or with this information, can the Corp assume my best move is to spend 0, so they can spend 1 instead of 2?

Knowing this, is my best move to spend 1 credit, hoping the Corp thinks this way, and tries to lock me out for 1 less credit, and hope for a tie?

Doesn't seem like it's worth the risk to the Corp to lose the game for 1 credit. Maybe Nash's theory does work in this case.

What do you think?
 
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M. Kern
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gar0u wrote:
What do you think?

I think analysing PSI Games is soooo 2015 ...
 
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Will H.
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Well, true.

I heard an interesting take on them that unless you don't play them, you'll never know if they impact your ability to win...
 
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Kim Choy
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gar0u wrote:
Not sure Nash Equilibrium can be applied to this, since it assumes there is a best move for both players. There is not really a fitness function to measure this. For example, in a game I just played:

1. Whoever scores the next agenda wins.

2. I was trying to steal an agenda, and had to deal with Caprice.

3. I only had 1 credit.

4. The Corp had plenty of money, so they would likely spend 2 credits to ensure I don't get access.

Now, is my best move to spend 0 credits, because I know the Corp is going to spend 2?

Or with this information, can the Corp assume my best move is to spend 0, so they can spend 1 instead of 2?

Knowing this, is my best move to spend 1 credit, hoping the Corp thinks this way, and tries to lock me out for 1 less credit, and hope for a tie?

Doesn't seem like it's worth the risk to the Corp to lose the game for 1 credit. Maybe Nash's theory does work in this case.

What do you think?

If you only have one credit and I, the corp, have enough money to bid 2 then I'm bidding 2 every time.
 
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umchoyka wrote:

If you only have one credit and I, the corp, have enough money to bid 2 then I'm bidding 2 every time.


For sure - if its 1 shot for the game, then there's no reason to ever bid 1 as corp. You'd only consider not bidding 2 if there was a chance that losing that $1 would:

a - prevent you from advancing the final agenda
b - prevent you from rezzing ice that you need to protect game winning agendas out of HQ / R&D.
 
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