January 2023

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Currently, Neural Networks are the most ubiquitous artificial intelligence method. Artificial intelligence (AI) has had massive breakthroughs over the last 15 years. From the year 2000 until 2008, computers were not able to get near human level performance in most pattern recognition tasks. For example, they had about 75% accuracy when converting spoken words to text, maybe a 30% chance of recognizing things in images (like people, planes, or mountains), and maybe a 90% accuracy rate on recognizing hand written text. Big data helped and fast computers helped, but they still were not getting human level performance. Different methods were used for each pattern recognition system. Now in 2023, neural nets seem are about as good as humans in many pattern recognition tasks and, in almost all cases, the computers are using neural nets to achieve human level performance.

Neural Nets were first discovered in the 1960s, but they were not very useful. They were used for some tasks in AI, but they were not the best tool for anything except maybe hand written digit recognition until about 2008. In about 2007 (or maybe 2006?), Geoffrey Hinton discovered how to train “deep” neural nets. Neural Nets have layers and it was very difficult to teach any neural net with more than three layers. Hinton found a way to train neural nets with up to 15 layers. (I attended the lecture at the NIPS conference in 2007 when Hinton introduced this new method–it was very exciting.) This method massively improved the pattern recognition abilities of neural nets and by 2010 neural nets were one of the best pattern recognition systems. In 2012, Hinton discovered the “Drop Out” training method. After Drop Out was introduced, neural nets became better than humans in many pattern recognition tasks like image recognition.

Over the last 10 years, three other big break throughs occurred: GANs (generative adversarial networks), transducers, and diffusion models. These breakthroughs have created neural nets that can create photo realistic faces, digital image art, and intelligently answer almost almost any question (the answer is correct only around 70% of the time). The biggest recent breakthrough is the GPT3 neural net which can pass final exams in several college subjects, write good essay on just about any topic, write short computer programs, and it can carry on a short conversation.

For further information, I am adding links for an introduction to neural nets that actually describes what they are, and a 17 page history of AI.



There are many games that have an investment phase followed by an exploitation phase including the card game Dominion, Terraforming Mars, Slay the Spire (during a combat), Saint Petersburg, Roll for the Galaxy, Master of Orion, Hansa Teutonica, Century Spice Road, Cash Flow and many more games.

One of the basic ideas is that you should not invest in something if there is not enough time for the investment to pay for itself.


Suppose that there are $T$ turns left in a game excluding the current turn, your current income level is I dollars per turn, and you have two choices:

  1. (invest) Invest this turn to increase your income from $I$ dollars per turn to $I +i$ dollars per turn. You would then receive $I+i$ dollars per turn for the remaining $T$ turns for a total of $(I+i)\cdot T$ dollars.
  2. (don’t invest) Receive $I$ dollars this turn and the remaining $T$ turns for a total of $(T+1)\cdot I$ dollars.



For example, assume that

  • it is currently turn 5,

  • at the start of turn 5 you have an income of $I$ = \$2 per turn,

  • the game ends on turn 9, and

  • you have the choice between two options:

  1. Invest this turn to increase your income from \$2 per turn to \$3 dollars per turn. You would then receive \$3 dollars per turn for turns 6,7,8, and 9 for a total of \$3 * 4 = \$12 dollars.

  2. Do not invest and you receive \$2 on this turn, turn 5, and the remaining turns 6,7,8, and 9, for a total of \$$2\cdot5$ = \$10 dollars.

For this example, $T=4$ because there are 4 remaining turns after the current turn. You have the option of increasing your income from \$2 to \$3, so the investment increases your income by $i$ = \$1 per turn.

The Correct Strategy

If you choose option 1, then your total earnings will be

invest_option_returns = $T\cdot (I+i)$ dollars.

If you choose option 2, then your total earnings will be

no_investment_option_returns = $(T+1)\cdot I$ dollars.

So, you should invest if

$$\begin{aligned}T\cdot (I+i) &> (T+1)\cdot I \\T\cdot I+T\cdot i &> T\cdot I + I \\T\cdot i &> I \\T&>\frac{I}{i}. \\\end{aligned}$$

So, according to the math above, you should not invest in something unless there are more than $$\frac{I}{i}= \frac1{r}$$ turns left in the game after the current turn where $$r=\frac{i}{I}$$ is sometimes called the return on investment (ROI).

When you invest $I$ dollars to get an increase of $i$ dollars per turn, it takes $I/i$ turns for the investment to “pay for itself”.


The Gain

If there are more than $\frac1{r}=\frac{I}{i}$ turns in the game, then you will gain

gain = invest_option_returns – no_investment_option_returns

$$\begin{aligned}&=T\cdot (I+i) – (T+1)\cdot I\\&=T I+T i – T i – I\\&= T i – I \\&= T r I – I \\&= (Tr -1) I \\&= I r  \left(T -\frac1{r}\right) \end{aligned}$$


dollars by choosing to invest.  The investment pays for itself after $1/r$ turns and every turn after that gives you $I r=i$ dollars.



For the example, $T=4, I=2,$ and $i=1$, so $r = i/I =0.5$. The correct strategy is to invest if

$$T > I/i,$$

but T=4 which is greater then I/i = 2, so the best option is the investment option.



I am hoping to write about applying this idea to several games over the next few weeks.