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annuity due formula

An annuity due would: have payments occur on January 1 of 2025, January 1 of 2026, and January 1 of 2027. Perform. You just need to discount cash flow to time zero. If we used the regular annuity formula or table, we would be given the future value of the above case as $610.51. If the policy continues to pay throughout the remainder of the annuitant’s life, it is called awhole life annuity. Annuity due. To calculate the payment for an annuity due, use 1 for the type argument. Examples. Step 2: $$i=8 \% / 1=8 \% ; N=1 \times 14=14$$ compounds Interest rate reduced while periods of time increase by frequency of compounding (m) i.e. The payments are made at the start of each period for n periods, and a discount rate i is applied. These payments are expected to be made on predetermined future dates and in predetermined amounts. Formula: Where, n = number of years R = Rate of return In the example shown, the formula in C11 is: = An annuity due is sometimes referred to as an immediate annuity.The future value of annuity due formula calculates the value at a future date. You will NEVER forget the difference between annuity and annuity due with this ONE WEIRD TRICK! Thanks to all of you who support me on Patreon. With an annuity due, payments are made at the beginning of the period, instead of the end. This formula can only be used in cases where the presentvalue of an investment is known. The payments are generally fixed and there are two values for an annuity, one would be future value, and another would be present value. An annuity due is a series of payments made at the beginning of each period in the series. In the example shown, the formula in F9 is: = The present value of an annuity due is the value of the annuity due in today’s dollars. Traditional notation uses a halo system where symbols are placed as superscript or subscript before or after the main letter. Formula. The value of the annuity due would be calculated on December 31, 2027. The calculation is usually made to decide if you should take a lump sum payment now, or to instead receive a series of cash payments in … Formula and Use. Annuity Calculator. The annuity formula to calculate the present value of an annuity due is: The annuity formula to calculate the future value of an annuity due is: Where, C = is the cash flow for the period, i = interest rate and n = number of years This can be expressed as follows: You da real mvps! What is the difference between the present value and future value of an annuity due? Future Value of Annuity Due Formula. It follows from the difference in an ordinary annuity and an annuity due that we can get the future value of an annuity due by growing the present value of an ordinary annuity with the same terms (periodic payment, periodic interest rate and total number of payments) over one more period. Annuity due differs from ordinary annuity in that periodic cash flows occur at the end of each period in an ordinary annuity. The present value of annuity due formula shows the value today of series of regular payments. The present value of an annuity due is used to derive the current value of a series of cash payments that are expected to be made on predetermined future dates and in predetermined amounts. Please note that these formulas work only on a payment date, not between payment dates. An annuity due’s future value is also higher than that of an ordinary annuity by a factor of one plus the periodic interest rate. Please note that these formulas work only on a payment date, not between payment dates. When using the formula, the discount rate (i) should be greater than the growth rate (g). Example notation using the halo system can be seen below. Annuity Due Formula What is the Formula for the Present Value of an Annuity Due? Others: Some other annuity types are fixed annuity and variable annuity. Actuarial notation is a shorthand method to allow actuaries to record mathematical formulas that deal with interest rates and life tables.. Work with the general annuity due. The rate of interest is 10%. The formula discounts the value of each payment back to its value at the start of period 1 (present value). Free annuity calculator to forecast the growth of an annuity with optional annual or monthly additions using either annuity due or immediate annuity. The last difference is on future value. A growing annuity due is sometimes referred to as an increasing annuity due or graduated annuity due. Whole life annuity-due- continued Current payment technique - continued The commonly used formula a x = X1 k=0 vk p k x is the so-calledcurrent payment techniquefor evaluating life annuities. This future value of an annuity due formula is an investigative tool that is used to estimate the total value of cash payments made at the beginning of a pay period. The formula for the future of value of an annuity due is derived by: Annuity Due can be defined as those payments which are required to be made at the start of each annuity period instead of the end of the period. i/m and n*m. Two methods for calculation The fact that the value of the annuity-due is greater makes sense because all the payments are being shifted back (closer to the start) by one period. Formula. The final value would be$3,133.94. Note that the PV of the ordinary annuity is 131.22 and the PV of the annuity-due is 140.40 (calculated as 131.22 x 1.07). $1 per month helps!! The following are the types of annuity: Ordinary Annuity: The payment or deposit of cash occurs at the year. An annuity-due is a type of retirement plan in which the same amount is invested each period and the interest rate remains fixed.. To calculate the present value of an annuity due, use the following formula: PV = C X {[1 - … An annuity is a series of payments made at equal intervals. These are the main formulas that are needed to work with annuities due cash flows (Definition/No Tutorial Yet). Let’s assume that someone invests$1,000 each year for 5 years at an annual interest rate of 7.5%. In other words, payments are made at the beginning of each period. Future Value of an Annuity Due: Let’s say that we want to calculate the future value of an annuity which pays $100 for 5 years and the payments begin at the beginning of the first period. The future value of an annuity due formula can also be used to determine the number of payments, the interest rate, and … PV: Stands for Present Value of Annuity PMT: Stands for the amount of each annuity payment r: Stands for the Interest Rate n: Stands for the number of periods in which payments are made The above formula pertains to the formula for ordinary annuity where the payments are due and made at the end of each month or at the end of each period. :) https://www.patreon.com/patrickjmt !! Excel Function So, there is a slight change in the formula for computing the future value. Annuity due payment with present value formula helps tocalculate each installment for a series of payments or cash flows where firstcash flow is instant. The formula discounts the value of each payment back to its value at the start of period 1 (present value). The most common example of an annuity due is the rent, as the payment should be made at the start of the new month. With an annuity due, payments are made at the beginning of the period, instead of the end. Immediate annuity calculator & ordinary annuity calculator to calculate annuity for ordinary annuity / immediate annuity, and annuity due. Future Value of an annuity due is used to determine the future value of a stream of equal payments where the payment occurs at the beginning of each period. The payments (deposits) may be made weekly, monthly, quarterly, yearly, or at any other regular interval of time. Finally substitute into the annuity payments Formula 11.1 to solve for Years. An annuity due differs from an ordinary annuity because the payments take place at the beginning of each year instead of at the end. 134 LIST OF FORMULAS Interest rate: r = d 1−dn Discount rate in terms of interest rate: d = r 1+rn Discount rate in terms of a bid: d = 360−3.6B n Compound … As in the case of an ordinary annuity, the present and future values of the annuity due are also calculated as first and last cash flows respectively. As such, calculations of an annuity due requires different formulas. Experiment with other retirement planning calculators, or explore hundreds of individual calculators addressing other … Annuity Calculator calculates the annual payout amount of an annuity. Since payments of the annuity due is made at the starting of each period. At the bottom of the page, an annuity formula can be found that shows how to calculate annuity. Annuity Due: The inflow or outflow of cash occurs at the beginning. Perpetuity: The annuity which is everlasting. Calculate the periodic interest rate ($$i$$, Formula 9.1) and the number of annuity payments ($$N$$, Formula 11.5 rearranged for $$N$$). An annuity due stops after a specific amount of time, a perpetuity goes on forever (in theory). The annual life annuity pays the annuitant (annuity policyholder) once each year as long as the annuitant is alive on the payment date. The future value of annuity due formula is used to calculate the ending value of a series of payments or cash flows where the first payment is received immediately. The Present Value of Annuity Due formula is used to calculate the present value of a series of cash flows, or periodic payments, that are generated by an investment in the future. The basic concept behind the present value of annuity due is the same as that of an (ordinary) annuity. Examples of annuities are regular deposits to a savings account, monthly home mortgage payments, monthly insurance payments and pension payments. To calculate present value for an annuity due, use 1 for the type argument. Therefore, the formula for the future value of an annuity due refers to the value on a specific future date of a series of periodic payments, where each payment is made at the beginning of a period. Recalling what distinguishes an annuity due from an ordinary annuity is the time of payments of the annuity. Annuity Due. Annuities can be classified by the frequency of payment dates. What is Annuity Due? Future value of annuity due (intra-year compounding) The value of annuity due at some future time evaluated at a given interest rate assuming that compounding take place more than one time in a year (Intra Year). Each cash flow is compounded for one additional period compared to an ordinary annuity. This calculator allows you to input payments that are made annually, semi-annually, quarterly, monthly, weekly or daily. 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Type of annuity due is sometimes referred to as an immediate annuity.The future of...


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Author: Subject: Microcontrollers & The Amateur Chemist
IrC
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http://www.ebay.com/itm/Mini-ATMEL-AVR-ATMEGA-STK500-USB-ISP...

Right now you can get a nice Mini ATMEL AVR ATMEGA STK500 USB ISP Programmer for almost $49 off normal price. I do not know how long the sale will last but I ordered one from this ebay seller while the price was this low. Looks like He has plenty in stock right now. "Science is the belief in the ignorance of the experts" Richard Feynman bfesser Resident Wikipedian Posts: 2114 Registered: 29-1-2008 Member Is Offline Mood: No Mood I'm betting that those are the counterfeit ones I've heard about. I've been struggling to get one of the SparkFun Pocket AVR Programmers  to work over the last couple weeks. Does anyone here have any experience with these? I've drawn the conclusion that purchasing one of these was a mistake. All I want to do with it at this point is flash updated code to the ATtiny2313  in my Geiger counter kit. I'm about ready to withdraw a little from my savings account and order a proper AVRISP mkII  [Edited on 18.9.13 by bfesser] arsphenamine Hazard to Others Posts: 236 Registered: 12-8-2010 Location: I smell horses, Maryland, USA Member Is Offline Mood: No Mood  Quote: Originally posted by bfesser I'm about ready to withdraw a little from my savings account and order a proper AVRISP mkII Get the AVRISP mkII at ~$35 from AVR, Mouser, Digikey, Newark, et.al.

The Arduino development environment supports it, as does the Windows Atmel Studio. Think of it as the gold standard for AVR embedded flashing.

The only thing you need to remember is that it will overwrite the bootloader in an otherwise serviceable Arduino.

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bfesser
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Rotary Telephone Pulse to DTMF with PIC

I've got another (unrelated) project going...

Yesterday, I picked up an old rotary dial  telephone (ITT 500, made in Canada) from the local surplus shop, which I plan to restore and gift to my father. After a little research, I've found that pulse dialing  is no longer supported by the service provider my parents subscribe to. [curses newfangled technology—everything's fucking digital now...]

I managed to find a neat little circuit  that I'd like to try out which will generate DTMF  signals from the dial's pulses using a PIC12F675. On the bright side, this'll be a great opportunity for me to learn to etch PCBs—something I've been meaning to do for years. On the down side, I've invested my time and funds (unsuccessfully) toward AVRs  up to this point, and have not worked with the PIC  µCs before.

Of course, there are many other ways I could go about converting the telephone, but none of them have the appeal and simplicity of this one. In addition, I'd like to preserve as much of the original circuitry and mechanics of the phone as possible, so replacing the guts with a RasPi or Arduino with direct VoIP  is out of the question.

My question is this: Should I bother trying to build and program one of these myself and risk miserable failure (again), or should I just wait until I can afford to buy a commercial adapter?

[I didn't have a question before writing this—just needed to rant to let out my frustration.]

watson.fawkes
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 Quote: Originally posted by bfesser I managed to find a neat little circuit that I'd like to try out which will generate DTMF signals from the dial's pulses using a PIC12F675.
That "neat little circuit" is nothing but a power supply and a crystal frequency reference (possibly not even needed) for the microcontroller chip. All the pulse detection and tone generation is done in software. Nothing an AVR can't do, and all the software development could be done on a stock Arduino.
watson.fawkes
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 Quote: Originally posted by IrC Mini ATMEL AVR ATMEGA STK500 USB ISP Programmer
 Quote: Originally posted by bfesser I'm betting that those are the counterfeit ones I've heard about.
And what would that be a counterfeit of, exactly? It's a flash programmer for all the devices listed. Smells more like your prejudice against the Chinese again, rather than anything about the device itself.
phlogiston
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Although some people can debate endlessly about the differences between AVR's or PIC's, the reality is that there is not all that much of a difference. There is a wide range of each series available with similar peripherals, for similar cost.
If you have invested in AVR tools and built up some experience with it, why change? Especially for a simple application like a DTMF generator, the simplest chips of either range can do the job, both at a cost of less than 1 Euro.
I use PICs mostly myself, primarily because I have grown familiar with their design and peculiarities. They are very cost effective. The microcontroller part of my circuits is usually less than Eur 2,-. I don't think the Arduinos come close to that yet, but I must admit I haven't checked anytime recently.

[Edited on 18-9-2013 by phlogiston]

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"If a rocket goes up, who cares where it comes down, that's not my concern said Wernher von Braun" - Tom Lehrer
watson.fawkes
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 Quote: Originally posted by phlogiston I use PICs mostly myself, primarily because I have grown familiar with their design and peculiarities. They are very cost effective. The microcontroller part of my circuits is usually less than Eur 2,-. I don't think the Arduinos come close to that yet, but I must admit I haven't checked anytime recently.
The small chips in the AVR series uses are about the same price, depending on chip features. The Arduino itself isn't nearly as cheap, but it's a development board, even if it's inexpensive enough for some folks just to embed one in their projects.
bfesser
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 Quote: Originally posted by watson.fawkes And what would that be a counterfeit of, exactly?
Of the genuine article. The one whose revenue goes to pay the salaries of the hard-working software and electrical engineers who designed and developed it—not to some corrupt Chinese elite who owns a factory shamelessly producing knockoffs and counterfeits  and violating copyrights and patents out of sheer greed.
 Quote: Smells more like your prejudice against the Chinese again, rather than anything about the device itself.
Don't try to start that bullshit argument again—not here. If you're so confident in the quality, why don't you purchase the one linked to with your own money, try it out, and get back to us (if and when it finally arrives in the post)?

 Quote: Originally posted by phlogiston If you have invested in AVR tools and built up some experience with it, why change?
Because I've invested in what are apparently defective and useless tools and my experience with AVRs so far (aside from the Arduino) has been anything but positive. Regardless, thank you for your input, phlogiston

watson.fawkes
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Quote: Originally posted by bfesser
 Quote: Smells more like your prejudice against the Chinese again, rather than anything about the device itself.
Don't try to start that bullshit argument again—not here.
I didn't start it, you started it by posting prejudiced comments. The only reason that you think it's bullshit is that you are apparently completely incapable of recognizing that you could be in error in any significant way.

 Quote: Originally posted by bfesser If you're so confident in the quality, why don't you purchase the one linked to with your own money, try it out, and get back to us (if and when it finally arrives in the post)?
I own one. It arrived just fine, albeit slowly, as all naval EMS packages do. I haven't had occasion to use it yet.

It's not a counterfeit, since it's not labelled as an Atmel product. And if you think that knockoffs are somehow wrong, you're discounting that the continuing dominance of the PC market started with predominantly US-based knockoffs of the IBM PC (such as Compaq). Atmel's not crying over the product I've got; it's got two AVR chips in it. And furthermore it promotes higher sales of AVR chips.

So do you have evidence that these clone devices have infringed copyrights of the board mask or the software? Do you have evidence that it wasn't independently developed? It's a simple circuit, and the AVR tool chain is open source. There's no inherent right to be able to keep innovation to yourself, and to think otherwise is to fight reality and lose.
arsphenamine
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 Quote: Originally posted by phlogiston Although some people can debate endlessly about the differences between AVR's or PIC's, the reality is that there is not all that much of a difference. There is a wide range of each series available with similar peripherals, for similar cost.
Speaking as someone who wrote device control software in assembly language for ~10 years, I beg to differ. I wanted to like the PIC but its haphazard architecture and overpriced development tools made it a non-starter back when it mattered.

The PIC I/O and its control are a disorganized collection of features that militate against any ready use. Even Intel's 80186 was better thought out. Don't get me started about PIC asm mnemonics. Any C language wrapper will be an ugly mess of special features just to get an ordinary job done. It's only a question of degree and how well the abstraction suggests the underlying hardware.

AVR's ATmegas are considerably easier to use, the tools are cheap or free (as in beer), and the learning curve is shallow and fast if you already know C.
bfesser
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Well, I used to code C++, and have had no problem with C for the Arduino. It sounds like I'd be better off focusing on the Atmel AVRs, rather than dividing my attention between the two. Thank you for your advice. I'll probably invest in a decent programmer soon, and hope to get one of my old computers running Linux to make things easier.

arsphenamine
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 Quote: Originally posted by bfesser Thank you for your advice. I'll probably invest in a decent programmer soon, and hope to get one of my old computers running Linux to make things easier.
If Windows owns your primary computer, download AVR Studio. There are online tutorials for programming the Arduino with it. If you have a USB port, you don't need a programmer ... yet.

When your Arduino prototype is done, you can embed the program on the \$20 Teensy which is the size of a commemorative stamp.

Plan B is to use the Coridium ARM7 board and its interpreted BASIC. Rapid prototyping gets you farther faster than MIPS.
bfesser
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I found a similar circuit+code for AVR:

Attachment: PulseToTone_2012_01_22.zip (414kB)

For completeness, here's the PIC one:

Attachment: pulse2dtmf2.zip (111kB)

[edit: typo]

[Edited on 20.9.13 by bfesser]

PeeWee2000
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Hey bfesser I bought the pocket avr programmer from sparkfun about a year back to attempt to program some attiny85 chips but never had any success with it so theres a good chance it is the programmer but I'd like to think not as sparkfun seems like a pretty reputable site to me but looks can be deceiving :/

And also I think having an arduino \ uC in the lab would be great, Id love to have a set up that measures the temperature at various points in my apparatus and automatically turns the hotplate off when it detects a temperature rise! Unfortunately I never really learned how to program so its been kind of a pipe dream but if other people more knowledgeable in programming helped out Id love to help make it happen. I do however have plenty of experience in electronic circuits so putting things together is no problem.

“Everything is relative in this world, where change alone endures.”

― Leon Trotsky
bfesser
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What timing! I didn't see your post until this morning, but I happened to be working on the AVR Pocket Programmer last night, and finally got it to work! Frustratingly, it turned out that the ribbon cable they supplied with the programmer was defective (open MOSI on the 6 pin plug). I won't get into how I figured this out, but I did manage to carefully disassemble the cable and re-wire it to work. Afterward, I spent a little time playing around with the programmer, and finally managed to flash some custom firmware onto blank ATTiny2313s for my Geiger counter.

I'm familiar with many programming languages, though I haven't kept up with it over the years (since changing majors from Computer Science to Chemistry). If there's any coding you'd like help with, just let me know and I'll be more than glad to have a look.

Tip: Type &mu; or &micro; for μ.

arsphenamine
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 Quote: Originally posted by bfesser I won't get into how I figured this out, but ...
It involved dogged persistence, a WTF moment, and a steady stream of creative profanity as you made the repairs, yes?
bfesser
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Yes. But it wasn't angry profanity, more relieved and joyful at finally getting the damned thing working. I plan to post on my modifications to the hardware and firmware of the Geiger counter soon (in another topic.

quantumchromodynamics
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hardware input and output

Specifically does anyone know of or have ideas about hardware that might attach to a drip valve of a separation funnel? A gear down stepper motor attachment? Is there an existing protocol for interconnecting heating mantles, stirrers, shaker tables, video reaction monitoring, etc? For me the software and the feedback control algorithms are the fun part. What seems missing are the sensors and the hardware to automate a reaction in glassware. RC airplane servo motors seem like a good start.
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 Quote: Originally posted by arsphenamine Speaking as someone who wrote device control software in assembly language for ~10 years, I beg to differ. I wanted to like the PIC but its haphazard architecture and overpriced development tools made it a non-starter back when it mattered. ...snip... AVR's ATmegas are considerably easier to use, the tools are cheap or free (as in beer).

Indeed, opinions differ, and, sure, it could be made a little easier, but that's a minor issue. It's not -that- difficult. Everyone who can program can figure out how to program PICs in any language they like best. Development tools are also available for free, who told you otherwise? I've also enjoyed writing assembly for various processors and microcontrollers for about 15 years now and found the PIC series to be quite useful and cost effective.

-----
"If a rocket goes up, who cares where it comes down, that's not my concern said Wernher von Braun" - Tom Lehrer
arsphenamine
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Quote: Originally posted by phlogiston
 Quote: Originally posted by arsphenamine I wanted to like the PIC but its haphazard architecture and overpriced development tools made it a non-starter back when it mattered. ...snip... AVR's ATmegas are considerably easier to use, the tools are cheap or free (as in beer).

Indeed, opinions differ, and, sure, it could be made a little easier, but that's a minor issue. It's not -that- difficult. Everyone who can program can figure out how to program PICs in any language they like best. Development tools are also available for free, who told you otherwise? I've also enjoyed writing assembly for various processors and microcontrollers for about 15 years now and found the PIC series to be quite useful and cost effective.
The operative words are "back when it mattered" which was early in the millenium . CCS development tools were the least expensive at the time, but still a few hundred dollars and the PIC guaranteed a steep learning curve. No thanks.

GNU toolchain propagation changed that in only five years, for PIC, AVR, and ARM alike, but Massimo Banzi's C++ wrapper for the Arduino made it a no-brainer. I'm uninclined to grovel over the entirety of an architecture any more when all I need is a small simple appliance.

I don't assert that the Arduino is without flaw, only sufficiently easier to get up and running than I found the PIC to be. I wonder if any of them has a readily accessible CTC (clock timer counter) mode. You can do it with arcane frobbage of the Arduino, but it is exactly that: arcane because the documentation is thin on that point.

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quantumchromodynamics, I have plenty of ideas, but it's more effort than it's worth. The laboratory glassware that we're all familiar with is designed to be manipulated by human hands. As far as I'm aware, when professional chemists want to automate things, they switch over to using purpose-built apparatus (Swagelok® valves/fittings, stainless steel reactors, PTFE-lined pumps, and the like). While I'm sure we could come up with solutions to handle our solutions in 'regular' glassware, it would likely be easier (and cheaper) to use the appropriate kit. I'm no expert on the engineering side of Chemical Engineering, so perhaps someone else could chime in here.

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http://www.ebay.com/itm/pH-Circuit-Sensor-for-Arduino-/20077...

Has anyone tried this PH board for the Arduino?

"Science is the belief in the ignorance of the experts" Richard Feynman
bfesser
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IrC, I linked to Atlas Scientific in the first post of this topic. I emailed them a few weeks back to see if they could sell me one of those boards without the epoxy coating or pin headers. Long explanation short; they said 'no.'

pH Circuit  (Atlas Sci.)
Arduino Shield
Raspberry Pi Carrier Board

IrC
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Been many months since I read this thread so I did not remember it but I still wonder if the board on ebay is worth buying. OK how about building a stirrer using an Arduino?

http://www.circuitsathome.com/mcu/programming/vigorius-stirr...

http://www.circuitsathome.com/mcu/programming/vigorius-stirr...

Also you would have extra lines to monitor and control temperature or maybe a color change in a reaction stopping (or starting) stirring or heat. Even controlling a dump of chemicals in a complex reaction where one may not wish to be standing near it. One could also dump a reaction into ice water if events get out of control. Using a temperature sensor, or the color recognition sensor (it could look for oxides of nitrogen due to their obvious color as example).

http://www.ebay.com/itm/1pcs-Color-Sensor-Color-Recognition-...

I just find it odd this is a fairly dead thread when the incredible number of uses abound for the Arduino (or other controller) as it relates to science. Maybe its just me but if I had access to the huge number of advances 50 years ago you people do today my entire lab would run itself while I watched old scifi reruns. Hell 50 years ago I didn't even have reruns.

OK I give your going to have to explain this as I have not had one to study yet:

"I emailed them a few weeks back to see if they could sell me one of those boards without the epoxy coating or pin headers. Long explanation short; they said 'no.'"

Looking at the pic I see it would take seconds to unsolder the headers right through the epoxy with my vacuum de-soldering head. I have had to do this many times in years past on waterproofed circuit boards. Examples are a metal detector and also HV supply boards for HeNe lasers. Often I had to remove and replace a part on the board, adding a clear epoxy coat after repair is done in the areas where I melted out the epoxy around the component pins.

Also wouldn't DCM remove the epoxy if one was very careful to not remove it in areas you did not wish to? I have done this on circuit boards hundreds of times in years past. Why do you need to remove it? Is it to get the board to fit somewhere else, and would a 90 degree bend on pins work if this was your reason? I can see why they would not, they likely have them manufactured in the Orient somewhere and cannot get boards mid production. Or they don't want you screwing with their design? Screwing with other peoples boards is not all that hard if this is their only reason.

One of these irons would remove the headers in seconds right through the epoxy without any harm to the board, leaving clean pads to solder wires or whatever it is you wanted to do. Just be careful near the chip resistor by the GND pin. All the others are out in the open and easy to do. After doing whatever it is you need to do, mix up a little clear epoxy and re-coat the areas where you melted the epoxy away. We're talking about a 15 minute job here no big deal.

[Edited on 10-23-2013 by IrC]

"Science is the belief in the ignorance of the experts" Richard Feynman
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# Crosstalk in concurrent repeated games impedes direct reciprocity and requires stronger levels of forgiveness

• Nature Communicationsvolume 9, Article number: 555 (2018)
• doi:10.1038/s41467-017-02721-8
Accepted:
Published online:

## Abstract

Direct reciprocity is a mechanism for cooperation among humans. Many of our daily interactions are repeated. We interact repeatedly with our family, friends, colleagues, members of the local and even global community. In the theory of repeated games, it is a tacit assumption that the various games that a person plays simultaneously have no effect on each other. Here we introduce a general framework that allows us to analyze “crosstalk” between a player’s concurrent games. In the presence of crosstalk, the action a person experiences in one game can alter the person’s decision in another. We find that crosstalk impedes the maintenance of cooperation and requires stronger levels of forgiveness. The magnitude of the effect depends on the population structure. In more densely connected social groups, crosstalk has a stronger effect. A harsh retaliator, such as Tit-for-Tat, is unable to counteract crosstalk. The crosstalk framework provides a unified interpretation of direct and upstream reciprocity in the context of repeated games.

## Introduction

Social dilemmas are situations where mutual cooperation is better than mutual defection and yet there is an incentive to defect1,2. Cooperation is normally opposed by natural selection unless mechanisms for the evolution of cooperation are in place3. One such mechanism is direct reciprocity, which is based on repeated interactions between the same two players4,5. In repeated social dilemmas, humans often learn to use adaptive rules, telling them when to cooperate, when to defect, and how to motivate others to cooperate6,7,8. Cooperation can be achieved if people adopt conditional cooperative strategies such as Tit-for-Tat5, Generous Tit-for-Tat9,10, or Win-stay, Lose-shift11,12. Conditional cooperation, paired with some amount of generosity, can maintain a healthy level of cooperation13,14,15,16,17,18,19,20,21. It can evolve even if initially rare22,23,24,25,26.

Most previous models of direct reciprocity (with a few notable exceptions27,28,29,30) have either assumed that (i) individuals only engage in one repeated game at a time or that (ii) an individual’s action in one game is independent of all its other interactions. Because humans often engage in many games simultaneously, the first assumption seems to be violated in most practical scenarios. Moreover, evidence from experimental studies suggests that also the second assumption of independence may not always apply31,32,33,34,35,36,37,38. We say that a player’s decision is subject to “crosstalk” when an interaction that a player has in one repeated game influences how the very same player behaves in another repeated game (Fig. 1a). For example, consider the interactions in a group of three individuals, “Alice”, “Bob”, and “Charlie” (Fig. 1b). Suppose that after a series of previous encounters, Bob is prompted for a decision whether to cooperate with Alice in the next round. In her last interaction with Bob, Alice has cooperated. Therefore, Bob who uses Tit-for-Tat, would now cooperate with Alice. But Bob’s last interaction had occurred with Charlie and in that interaction Charlie had defected. Crosstalk now means there is some chance that Bob defects with Alice although direct reciprocity would mandate Bob to cooperate. Bob’s state with respect to Charlie influences his decision with respect to Alice.

Such crosstalk can result from various psychological processes. For example, experiments on upstream reciprocity suggest that subjects who have received help in their previous interaction often consciously choose to “pay it forward”22,31,32,33. Alternatively, crosstalk may also occur when subjects have limited working memory34,35,36. In that case, subjects may confuse their co-players’ past actions, which may in turn lead them to reward the wrong person for past cooperative behaviors. We propose a mathematical framework that allows us to quantify how crosstalk affects the cooperation dynamics within a population. We show that, in the presence of crosstalk, a single defector can lead to the complete breakdown of cooperation in an arbitrarily large group of conditional cooperators. Nevertheless, cooperation can prevail if the population is structured and if subjects are sufficiently forgiving. For our model, we do not need to specify the particular psychological process at work: the resulting behavioral dynamics are independent of whether crosstalk is the result of a conscious decision (as in upstream reciprocity), or the consequence of a subconscious error (as when individuals confuse the past actions of their co-players). However, the interpretation of our results will often depend on the specific psychological mechanism that gives rise to crosstalk. We revisit this matter in the “Discussion” section.

## Results

### Framework for crosstalk between concurrent repeated games

We consider a group of N individuals. Each individual plays a pairwise repeated prisoner’s dilemma (PD) with each interaction partner. These repeated games occur concurrently. At each time step, we choose a random pair of players for a single interaction (Fig. 1b). Each player uses a reactive strategy, defined by two parameters, p and q, which denote the probability to cooperate if the same co-player in the previous round has either cooperated or defected, respectively. The class of reactive strategies includes many well-known examples, such as always-cooperate (ALLC), always-defect (ALLD), tit-for-tat (TFT: Supplementary Fig. 1b), and Generous Tit-for-Tat10 (GTFT: Supplementary Fig. 1d). Reactive strategies can be implemented by stochastic two-state automata39,40,41,42. The two states are labeled C and D (see Supplementary Fig. 1a). In the next interaction, a player cooperates if she is in state C and defects if she is in state D. Cooperators pay a cost c for their co-player to receive a benefit b>c. Defectors do not incur a cost and their co-player does not receive a benefit. The player’s strategy determines how the player’s state is updated after an interaction has taken place.

In our setup, each player uses a specific strategy for all of her interactions, but has distinct automata to hold the games with all of her different co-players in memory (Supplementary Fig. 2). For example, a player using TFT can be in different states (C or D) with different co-players, but uses the same strategy to update her states against all of her co-players. The separate automata enable players to remember previous interactions and to react in future rounds according to their respective history with each co-player.

Crosstalk between two repeated games occurs if a player’s state with respect to one interaction partner displaces the player’s state with respect to another player (Supplementary Fig. 2). Specifically, we assume that, before each interaction, there is a probability γ that the players’ state with respect to the current co-player is replaced by the state with respect to the previous co-player (other variants of crosstalk will be discussed below). The crosstalk rate γ [0, 1] specifies how often crosstalk occurs. In the special case of no crosstalk, γ = 0, players perfectly distinguish between all their opponents, and we recover the scenario considered in previous studies of direct reciprocity2,5. For positive crosstalk rates, cooperative and defective behaviors can cascade in the players’ social network: a player’s action in one game can affect how the co-player acts in a different game, which in turn may influence again other games (Fig. 1). Therefore, crosstalk causes ripples that propagate in social networks. The overall effect of crosstalk depends on the structure of the population. We represent this structure by arranging players on a graph43,44,45, where edges between players denote interactions (Fig. 2; “Methods” section). While our framework is applicable to arbitrary population structures, we illustrate the effects of crosstalk using four regular networks (Fig. 3a), ranging from a circle (where each player has exactly two interaction partners) to the complete graph (where all players interact with everyone else).

### Cooperative and defective behavior spreads across population

We utilize stochastic computer simulations to study the cooperation frequency in a population over time, and derive mathematical recursions to calculate the long-run payoffs in the steady state (“Methods” section). To illustrate how crosstalk leads to the spread of defection in a generally cooperative society, we place a single ALLD player in a network of N − 1 conditionally cooperative players (Figs. 2 and 3a). When the conditionally cooperative players use TFT (which is given by p = 1, q = 0) and crosstalk occurs, γ > 0, the ALLD player can turn all remaining players into defectors eventually (Fig. 2a), independent of the population structure and the crosstalk rate (Supplementary Fig. 3). The spread of defection can be prevented if the cooperative players use more generous strategies, with p = 1 and 0 < q < 1. We refer to such strategies as GTFT. The impact of different q values will be discussed below. At first, we choose q = 1/3. If the single ALLD player is placed among GTFT players, cooperation frequencies converge to a positive value (Fig. 2b), with the eventual equilibrium rates depending on the population structure and on the crosstalk rate (Supplementary Fig. 3). We can also observe the opposite effect: a single ALLC player can increase the cooperation rates in a population of stochastic TFT players using p = 1 − $ϵ$ and q = $ϵ$ (Supplementary Fig. 4c).

Comparing the effect of different population structures, we find that a GTFT population can maintain cooperation more easily if players are arranged on a circle instead of a complete graph (Fig. 3a, b, Supplementary Fig. 5). For a population size of N = 16, the cooperators obtain a higher average payoff than the defector for crosstalk rates up to γ = 0.85 on a circle, and for up to γ = 0.41 on a complete graph. In networks with a low degree, players are more likely to give the adequate response with respect to their current co-player, because, if crosstalk occurs, the current co-player is more likely to coincide with the previous co-player such that crosstalk becomes inconsequential. For this reason, all other explored population structures exhibit crosstalk thresholds between the circle and the complete graph (Fig. 3b).

To investigate the recovery properties after a mistake, we computed the amount of time that a population of conditional cooperators with strategy (1, q) and q > 0 needs to return to full cooperation after a single defection event. We find that crosstalk leads to a significantly faster recovery (Supplementary Fig. 6). Intuitively, when the crosstalk rate is high, a player’s automata are updated more frequently (once before the interaction takes place, and once after the interaction). Because the players apply strategies with p = 1 and q > 0, each updating event is biased towards increasing cooperation: every cooperative act puts the co-player into a cooperative state, whereas defective acts are forgiven with probability q. Moreover, we show that the recovery time is monotonically increasing with the average degree (k) of the population structure and monotonically decreasing with the probability to cooperate after defection (q; see Supplementary Note 1 for more details).

### Crosstalk requires the right level of forgiveness

We calculate the generosity parameter q of GTFT to optimally cope with crosstalk. To this end, we consider two different optimality criteria. First, we calculate the most generous strategy that is able to resist invasion by a single ALLD player. That is, for a fixed population structure and a given crosstalk rate, we derive the reactive strategy (1, qM) with maximum qM such that N − 1 players with this strategy get at least the same average payoff as the single defector. Analytical calculations for the complete graph and numerical results for all other population structures show that higher crosstalk rates and higher network degrees (that is a higher number of neighbors) require the cooperative players to be less generous (Fig. 3c, “Methods” section). For the second optimality criterion, we calculate the cooperative strategy with the most robust level of generosity qR such that N − 1 players with strategy (1, qR) have the highest relative payoff advantage compared to the single ALLD player. The most robust level of generosity exhibits a non-monotonic dependence on the crosstalk rate (Fig. 3d). In the absence of crosstalk, γ = 0, the perfectly reciprocal TFT strategy is most robust against invasion of ALLD. As the crosstalk rate increases, the most robust level of generosity qR first increases, but then decreases again. Intuitively, for the robustness of a conditionally cooperative population against ALLD, high values of the generosity parameter q have two opposing effects. On one hand, high values of q make it less likely that the defectors’ actions propagate through the network. On the other hand, high values of q also let the players be more forgiving against the defector, and hence increase the payoff of the ALLD player. When crosstalk is rare, conditional cooperators can prevent the spread of defection by choosing a small value for q. Once crosstalk is sufficiently frequent, however, players can no longer fully prevent defection from spreading. Instead, they rather need to keep the defector’s payoff low, by choosing a smaller q value. These results confirm that in the presence of crosstalk, γ > 0, subjects should show some amount of generosity (q > 0), but not too much; we find q < qM, with qM depending on the crosstalk rate and on the population structure. Only for the circle, cooperation can prevail even when crosstalk is abundant (Fig. 3d).

The above analysis is based on a comparison between conditional cooperators and a specific invader, ALLD. More generally, we find that conditionally cooperative strategies (1, q) with q < qM in fact resist invasion by all possible invading strategies (pʹ, qʹ) for the complete graph. This analysis also reveals that there are three classes of strategies in total that are stable against arbitrary invaders (Supplementary Note 1). The first class consists of the conditionally cooperative strategies just described. The second class consists of uncooperative strategies (p, 0), with p sufficiently small (see Supplementary Note 1 for the exact condition). In particular, this class contains ALLD. When adopted by all players in a population, strategies of this class eventually lead to full defection. Finally, the third class consists of strategies (p, q) analogous to equalizer strategies of direct reciprocity46,47,48,49. When applied by all residents in the population, equalizer strategies guarantee that the payoff of a single invader is independent of the invader’s strategy.

### Crosstalk impedes the evolution of cooperation

These results raise the question to which extent subjects themselves would learn to apply cooperative strategies with a sustainable degree of generosity. To explore that question, we implemented a simple model of cultural evolution where players are allowed to adopt new strategies over time, based on their current strategy’s success (“Methods” section). According to this process, strategies that yield a comparably high payoff are more likely to be imitated by other players50,51. In addition, players may occasionally also experiment with new stochastic strategies, which introduces novel behaviors into the population. These two events, imitation and exploration, take the role of selection and mutation in models of biological evolution. We show that a birth-death process as used in many biological applications yields the same results (Supplementary Note 1). We simulated the evolutionary dynamics for various population structures and crosstalk rates, assuming that experimentation events are relatively rare23. In the plane of reactive strategies, we observe that for most of the time, players either apply defective strategies with p ≈ q ≈ 0, or cooperative strategies with p ≈ 1 and 0 < q < qM (Fig. 4a–c). None of these strategies are evolutionarily stable52. Instead, when residents apply one of these strategies, neutral or nearly neutral mutants can often invade and pave the way for mutants of another strategy class42. The relative weight of these two strategy classes depends on the crosstalk rate and on the population structure (Fig. 4d). While cooperative strategies readily evolve on the cycle even for substantial crosstalk rates, they become less abundant as the population structure changes to a square lattice, or to a complete graph.

To understand the effect of crosstalk in more detail, we analyzed how easily other strategies fix in a resident population that either consists of ALLD players or GTFT players (Supplementary Fig. 7). Without crosstalk, GTFT is much more successful in resisting mutant invasions. On average, more mutants need to be introduced until the first mutant fixes. Moreover, successful mutants typically have a strategy that is very similar to GTFT (whereas ALLD is typically invaded by TFT-like strategies rather quickly). However, as the crosstalk rate increases, the invasion time into GTFT drops considerably, and successful mutants no longer need to be cooperative themselves.

Interestingly, we find that crosstalk favors the stability of extortionate strategies. With an extortionate strategy, players can guarantee that they never get a lower payoff than their co-player, while simultaneously acting such that it is in the co-player’s best interest to cooperate unconditionally47. In classical models of direct reciprocity, extortionate strategies are unstable and they can only succeed if the population size is small18,19,53. In contrast, extortionate strategies can thrive even in large populations when crosstalk is sufficiently abundant (Supplementary Fig. 8). However, this success comes at a cost. By becoming stable, extortionate strategies lose one of their most appealing properties: when crosstalk is abundant, a rare invader in an extortionate population does no longer benefit from being unconditionally cooperative. Instead, the best response is to be extortionate as well (see Supplementary Note 1 for details).

To analytically understand the impact of crosstalk on the evolutionary dynamics, we explored a deterministic model of evolution in well-mixed populations54,55,56. The singular strategies of these dynamics consist exactly of the three strategy sets that resist invasion by rare mutants that we described in the previous section: conditional cooperators, defectors, and equalizers. Which of these strategy classes is reached in the course of evolution now depends on the initial population (Supplementary Fig. 9a). As the crosstalk rate increases, the number of initial populations that eventually end up in a conditionally cooperative state decreases (Supplementary Fig. 9b). In this deterministic model, crosstalk thus acts by reducing the basin of attraction of the cooperative equilibria.

### Alternative models of crosstalk

So far, we have analyzed one particular model of crosstalk: when prompted for the next decision, a player instead reacts to her most recent interaction, and this interaction could have been with someone else. But other implementations are conceivable.

For example, the memory state that a player holds for her current co-player could be replaced by the memory state of a random co-player, who may not coincide with the current or the most recent co-player. In this case, when deciding what to do for the next round in a particular game, the player uses with probability γ the state of a random game that she is holding in memory. We find that this alternative implementation of crosstalk differs in its short-term dynamics but converges to the same steady state as our original model (Supplementary Fig. 10). Thus, Figs. 3 and 4 immediately apply to this type of crosstalk as well.

Alternatively, a player’s decision could depend not only on the previous interaction with one particular opponent. Instead, the player might consider an average across her recent experiences with all her co-players. We therefore introduce aggregate reactive strategies (Supplementary Note 1). Players using these strategies compute a weighted cooperation score across all their co-players. This cooperation score incorporates the last action of the current co-player with weight 1 − γ and the average cooperation rate across all co-players’ last actions with weight γ. The obtained score is then compared to an exogenous cooperation threshold τ. Players cooperate with probability p if the weighted cooperation score exceeds τ; otherwise they cooperate with probability q. In particular, if the crosstalk rate γ is zero, these strategies again correspond to the classical reactive strategies of direct reciprocity2,10. We explore the effects of γ and τ by considering a single defector in a population of conditional cooperators. The conditional cooperators apply a strategy with p = 1, q = 1/3 and some value τ, a strategy to which we refer to as Aggregate Generous Tit-for-Tat (AGTFT). While the cooperation dynamics of this model are qualitatively different from our previous results, we again find that higher crosstalk rates impede the stability of cooperation across all population structures (Supplementary Figs. 11 and 12).

## Discussion

Classical models of direct reciprocity require that players provide a targeted response to each of their co-players2,5. Experimental results and everyday experience indicate that players’ decisions can be affected by unrelated events that occur in their interactions with others31,32,33,34,35,36,37,38. Crosstalk arises when a player has simultaneous repeated interactions with several opponents. Importantly, crosstalk is different from previous approaches that combined direct and indirect (downstream) reciprocity. In models of downstream reciprocity, a player’s strategy depends on the co-player’s reputation, and hence on the co-players’ interactions with others38,57,58,59,60,61,62,63,64. A combination of direct reciprocity and downstream reciprocity can promote cooperation because a single defection in one game may lead several unaffected co-players to retaliate against the defector27,28,29. However, downstream reciprocity makes stronger assumptions on the information players have when making their decision. It requires a player to observe other players’ interactions or reputations to respond accordingly. In contrast, crosstalk is a much more elementary mechanism. It occurs “within” each player and does not rely on additional external information about independent interactions of unrelated players. Our notion of crosstalk is general: it captures that a player’s decision in one game can be affected by the player’s previous experience in another game, but it does not depend on the psychological process responsible for this interdependency.

Depending on the specific process at work, crosstalk is amendable to different interpretations. Our framework can be taken as a model of upstream reciprocity in the context of repeated games. Under this interpretation, cooperation or defection received from one person is sometimes consciously “paid forward” to another person. Previous analytical models have either focused on direct reciprocity or on upstream reciprocity separately2,65,66. The framework of crosstalk allows us to explore the consequences when both modes of reciprocity act simultaneously, and possibly interfere with each other. We recover previous results65,66,67 that upstream reciprocity alone is most likely to yield cooperation when the population is highly structured. However, our results suggest that cooperation can even be maintained in well-mixed populations when upstream reciprocity is sufficiently coupled with direct reciprocity (i.e., when the crosstalk rate γ is sufficiently small).

Alternatively, crosstalk can serve as a model of individuals with limited working memory. According to this interpretation, crosstalk occurs when individuals confuse their various co-players, which introduces a type of behavioral noise into the cooperation dynamics. This noise is different from simple implementation errors considered in previous models2,9,18,19,36,68,69. Implementation errors only affect the repeated game in which they occur (Supplementary Fig. 4a, b). But crosstalk spreads from one game to another and therefore through the population (Supplementary Fig. 4). Only in the presence of crosstalk, a single defector can turn a whole population of TFT players into defectors.

We consider upstream reciprocity and confusion as different psychological processes which independently can give rise to crosstalk. Although these two processes are subject to different interpretations, according to the above discussed implementation, they lead to the same cooperation dynamics within a population. High degrees of upstream reciprocity, just as high degrees of confusion, undermine the ability of direct reciprocity to sustain cooperation.

Crosstalk provides a general framework with applications beyond the examples studied here. Future models could explore, for example, crosstalk between independent games that differ in their payoff structure70, or when subjects engage in simultaneous games that involve more than two players. Similarly, one may study interactions in which the crosstalk rate itself depends on exogenous parameters, such as the number of neighbors, or the benefit of cooperation. Finally, we explored a model in which players aggregate across their last experience with all co-players. Further generalizations are conceivable. For example, players may defect against all their co-players as long as at least one of their automata is in the D state, or they may remember more than their co-player’s last action26,71. Under crosstalk, the players’ individual games are no longer considered in isolation, but they are embedded into the context of all concurrently ongoing interactions. Crosstalk requires stronger mechanisms for forgiveness especially in a more highly connected world. A harsh retaliator such as Tit-for-Tat is particularly unable to deal with crosstalk. This is an interesting message for our current society.

## Methods

### Computer simulations

To simulate the effect of crosstalk on the cooperation dynamics among players with fixed strategies, we consider a population of size N playing a repeated Prisoner’s Dilemma (PD). The population is given by a graph where the nodes represent the players, and the edges reflect all possible interactions between players. Only players connected by an edge can be paired to play the PD. Players use separate two-state automata for each of their neighbors on the interaction graph41. The two states of each automaton are labeled C (cooperation) and D (defection). These states are updated according to the player’s reactive strategy (p, q), see Supplementary Figs. 1 and 2. The parameter p denotes the probability to move to state C if the co-player has cooperated in the previous game (whereas the complementary probability 1 − p gives the likelihood to move to state D). Similarly, q denotes the probability to move to state C if the co-player has defected (and 1 − q is the respective probability to move to state D).

In each round, an edge of the interaction graph is chosen uniformly at random. A single PD is played among the two players adjacent to the chosen edge. With probability 1 − γ a player acts according to the respective automaton state associated with this co-player; with probability γ crosstalk occurs and the player refers to the state of the automaton updated in her last interaction instead. After the game, the automata states are updated according to the game outcome and the players’ strategies. This elementary step is then iterated for a large number of rounds. For the simulation results depicted in Supplementary Fig. 5, we simulated 4000 games per realization (on average 500 games per player) and averaged across 104 realizations to obtain the stationary payoff of GTFT and ALLD players for a given population structure and crosstalk rate.

So far, we assumed that every edge in the interaction graph is chosen with the same probability. However, some interactions can occur with a higher frequency than others. To investigate the effects of different interaction frequencies, we studied the spread of defective behavior in a population of GTFT players populating a 5 × 5 lattice (Supplementary Fig. 13). We increased the interaction probability of all players on the central horizontal line by 10-fold (see orange edges) and observe how defective behavior spreads much faster along the horizontal axis than along the vertical axis. Within the analytical framework, interaction probabilities w ij are given by the connectivity matrix W (see next section for details).

In the second studied type of crosstalk, again with probability γ crosstalk occurs and the player refers to the state of a random automaton, chosen from all her interaction partners with equal probability (Supplementary Figs. 2 and 10).

### Analytical derivation of steady-state payoffs

To derive an explicit representation for the payoffs that players receive in the long run, we suppose that each player i adopts some fixed reactive strategy (p i , q i ). The population structure is given by an N × N connectivity matrix W = (w ij ). The entries w ij give the probability that the next interaction in that population occurs between players i and j. In particular, the connectivity matrix W is symmetric w ij  = w ji , and satisfies w ii  = 0 and $∑ i < j w i j =1$. As in the computer simulations, we focus on networks in which each link is played with equal probability. That is, if player i and j are connected, then w ij  = w for some constant w > 0 that depends on the network structure, but is independent of the players i and j. For example, because there are N(N − 1) / 2 different links in a complete graph, well-mixed populations can be represented by a connectivity matrix with w ij  = 2 / (N(N − 1)) for all i ≠ j. As another example, populations on a cycle are represented by w ij  = 1/N if $i$ and j occupy neighboring sites, and w ij  = 0 for all other i, j.

Let $w ̄ i = ∑ j = 1 N w i j$ denote the probability that the next interaction in the population involves player i. Moreover, let $y i j t$ be the probability that player i is in state C against player j at time t, and let $y i t$ be the probability that player i is in state C with respect to her previous co-player. We can calculate $y i , j t + 1$ using the following recursion

$y i j t + 1 = 1 - w i j y i j t ⏟ If player i did not interact with player j in previous round + ( 1 - γ ) w i j ⋅ y j i t p i + 1 - y j i t q i ⏟ Players i and j interacted , and player j ′ s action was not subject to crosstalk + γ w i j ⋅ y j t p i + 1 - y j t q i ⏟ Players i and j interacted , and player j ′ s action was subject to crosstalk .$
(1)

To calculate the player’s long run cooperation frequencies, we note that in the steady state, cooperation rates are independent of the time period, and hence $y i j t + 1 = y i j t =: y i j$. Moreover, because interactions are fully random, the stationary probability y i that player i is in state C with respect to her previous co-player simplifies to $y i = ∑ k = 1 N w i k w ̄ i y i k$, which is the weighted average that player i is in state C with respect to a random co-player. In that case, we can rewrite Eq. (1) as

$y i j - ( 1 - γ ) p i - q i y j i -γ p i - q i y j = q i .$
(2)

The Eqs. (2) represent a system of N(N − 1) linear equations in the unknowns y ij with i ≠ j. By solving this inhomogeneous system, we can calculate the stationary frequency $ŷ i j$ to find player i in state C with respect to player j. Given the stationary frequencies $ŷ i j$, we can calculate the payoff π i of player i by averaging over all co-players,

$π i = ∑ j = 1 N w i j w ̄ i ⋅ ( 1 - γ ) ŷ j i + γ ŷ j ⋅ b - ŷ i j ⋅ c .$
(3)

We note that this method applies to general crosstalk rates γ, general population structures, and general population compositions (e.g., populations with more than two different strategies present). As shown in Supplementary Fig. 5, these analytically derived payoffs are in excellent agreement with the computer simulations. In the Supplementary Note 1, we show how Eqs. (2) and (3) can be further simplified for well-mixed populations. In that case, we can also provide explicit expressions for how generous cooperative strategies of the form (1, q) are allowed to be to resist invasion of ALLD (as depicted in Fig. 3c, d).

### Setup of the evolutionary simulations

To explore the evolution of strategies under crosstalk, we consider a simple model of cultural evolution, the pairwise comparison process50,51. As common in studies on the evolution of strategies in repeated games10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,40,41, we assume a separation of time scales: the time it takes individuals to play their repeated games is short compared to the evolutionary timescale at which individuals adopt new strategies. This assumption allows us to use the players’ stationary payoffs, as given by Eq. (3), when simulating the evolutionary trajectory of a population.

For the evolutionary simulations, we consider a population with fixed population structure and fixed crosstalk rate γ. In each evolutionary time step, there are two possible events, imitation or random strategy exploration. To model imitation events, we assume that two individuals are randomly drawn from the population. We refer to these two individuals as the “learner” and the “role model”, respectively. Herein, we aim to compare the effect of crosstalk across different population structures. To allow for a fair comparison, we assume that, while payoffs are calculated for the given population structure, strategy updating occurs globally. As a consequence, the learner and the role model do not need to be neighbors in the direct interaction network. With this assumption, we rule out the formation of cooperative clusters, which would additionally favor the evolution of cooperation in networks with a low degree44. After selecting the learner and the role model, their payoffs πL and πR are calculated according to Eq. (3). We assume that the learner adopts the role model’s strategy with probability ρ = [1 + exp(−s(πR − πL))]−1. The parameter s ≥ 0 measures the strength of selection. When selection is weak, $s≪1$, payoffs are largely irrelevant for imitation and the imitation probability approaches 1/2, irrespective of the players’ strategies. When selection is strong, $s≫1$, players tend to adopt only those strategies that yield a higher payoff than their own strategy. In addition to these imitation events, we allow for random strategy exploration. When such an exploration event occurs, one player is randomly drawn from the population. This player then adopts a new strategy (p, q), which is uniformly drawn from all reactive strategies. Following the approach of Imhof and Nowak23, we assume that these exploration events are rare. As a consequence, the population is homogeneous most of the time. Only occasionally, a mutant strategy enters the population due to random strategy exploration. This mutant strategy than either goes extinct or fixes before the next exploration event occurs. By simulating this process over a long timespan, we can record how often the population applies certain strategies (p, q), and we can compute the resulting average cooperation rate over an evolutionary timescale. In Fig. 4, we show corresponding results for the cycle, the square lattice, and for the complete graph, assuming parameter values of population size N = 16, benefit b = 10, cost c = 1, and selection strength s = 1. Other parameter values lead to qualitatively similar results, provided that selection is sufficiently strong and that the benefit of cooperation is sufficiently high to allow for the evolution of cooperation. In the Supplementary Note 1, we show that analogous results apply when we consider a birth-death process instead of the pairwise imitation process considered herein.

### Data availability

No data sets were generated during this study.

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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## Acknowledgements

This work was supported by the European Research Council (ERC) start grant 279307: Graph Games (C.K.), Austrian Science Fund (FWF) grant no P23499-N23 (C.K.), FWF NFN grant no S11407-N23 RiSE/SHiNE (C.K.), Office of Naval Research grant N00014-16-1-2914 (M.A.N.), National Cancer Institute grant CA179991 (M.A.N.) and by the John Templeton Foundation. J.G.R. is supported by an Erwin Schrödinger fellowship (Austrian Science Fund FWF J-3996). C.H. acknowledges generous support from the ISTFELLOW program. The Program for Evolutionary Dynamics is supported in part by a gift from B Wu and Eric Larson.

## Author information

### Author notes

• Johannes G. Reiter

Present address: Canary Center for Cancer Early Detection, Department of Radiology, Stanford University School of Medicine, Palo Alto, CA, 94304, USA

1. Johannes G. Reiter and Christian Hilbe contributed equally to this work.

### Affiliations

1. #### Program for Evolutionary Dynamics, Harvard University, Cambridge, MA, 02138, USA

• Johannes G. Reiter
•  & Martin A. Nowak
2. #### IST Austria (Institute of Science and Technology Austria), Klosterneuburg, 3400, Austria

• Christian Hilbe
•  & Krishnendu Chatterjee
3. #### Yale Institute for Network Science and Department of Psychology and Department of Economics, Yale University, New Haven, CT, 06520, USA

• David G. Rand
4. #### Department of Organismic and Evolutionary Biology and Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA

• Martin A. Nowak

### Contributions

J.G.R., C.H., D.G.R., K.C., and M.A.N.: Designed the research; J.G.R., C.H., D.G.R., K.C., and M.A.N.: Performed the research; J.G.R., C.H., D.G.R., K.C., and M.A.N.: Wrote the paper.

### Competing interests

The authors declare no competing financial interests.

### Corresponding author

Correspondence to Martin A. Nowak.


text:
# Neutron stars: the equation of state, superconductivity/superfluidity and transport coefficients (PHAROS WG1+WG2 meeting)

Sep 26 – 28, 2018
Department of Physics (University of Coimbra)
Europe/Lisbon timezone

## Effects of the tetraneutron condensation in neutron stars

Sep 26, 2018, 5:30 PM
20m
Room E10A

### Speaker

Dr Oleksii Ivanytskyi (University of Salamanca)

### Description

Based on recent experimental and theoretical hints on possible formation of a resonant four-neutron system we study effects of appearance of such a cluster in neutron rich baryon matter inside NSs. For this purpose we employ a relativistic mean field approach which includes nucleons, $\Delta$-baryons as well as light nuclear clusters. Our analysis demonstrates that tetraneutrons existing as the Bose-Einstein condensate can affect the equation of state of cold baryonic matter and observable characteristics of neutrons stars. Tetraneutron driven suppression of $\Delta$-baryons is another important result of our study. Influence of tetraneutrons on formation of superconducting phase is also discussed.

### Primary authors

Dr Oleksii Ivanytskyi (University of Salamanca) Prof. Conrado Albertus Torres (University of Salamanca) M. Angeles Perez-Garcia (University of Salamanca and IUFFyM)


text:
Find the prime factorization in the DLP

Suppose we have $g^x \equiv h \pmod N$ Where $N = pq$ and $p$, $q$ are distinct primes. Also $(p-1)/2$ and $(q-1)/2$ are primes.

If we know what $x$, $g$ and $h$ are, is it possible for us to know what $p$ and $q$ are? I'm trying to find out but I'm running into closed doors.
I tried the following:
$gcd(p-1 , q-1) = 2$
$g^{(p-1)(q-1)/2} \equiv 1 \pmod {pq}$
so $1 \equiv h^{(p-1)(q-1)/2} \pmod {pq}$

Any help is appreciated.

• For the question "If we know what $x$, $g$ and $h$ are, is it possible for us to know what $p$ and $q$ are?", I doubt there is a positive answer $\;$ However another interesting question is: "If we knew a method to find $x$ such that $g^x\equiv h\pmod N$ given $(g,h)$, would it be possible for us to know what $p$ and $q$ are?" – fgrieu Mar 5 '15 at 7:49

2 Answers

As a hint, suppose we pick a random $h$, and compute $g = h^3 \bmod N$. We then find the minimal value $x$ where $g^x \equiv h \pmod{N}$ (and, yes, such an $x$ will exist, assuming $p, q > 7$).

What can we deduce from such a value of $x$?

• For this to work, $x$ must be $-3 \pmod {\phi (N)}$ right? – mike russel Mar 4 '15 at 14:41
• @mikerussel: Actually, $3^{-1}$, not $-3$. – poncho Mar 4 '15 at 14:45
• Okay so how does this help us retrieve $p$ and $q$? – mike russel Mar 4 '15 at 14:45
• @mikerussel: What is $3x-1 \bmod {(p-1)(q-1)/2}$ with good probability? – poncho Mar 4 '15 at 14:59
• We know $x = 3^{-1}$ so $3x-1= 0$ So $(p-1)(q-1)/2$ divides $3x-1$. – mike russel Mar 4 '15 at 15:02

Just knowing any such triple would not help at all. Reason:

Choose $g$ and $x$ arbitrarily, calculate $h$. Now you got such a triple, but it shouldn't help you factorize $N$, unless you have another interesting property.

If on the other hand you can somehow calculate roots (given $x$ and $h$, and then find $g$), then you can break RSA: Choose $x=e$, and your root algorithm finds the plaintext to a ciphertext.

If you can find square roots (and there are always 4 of them with $n=pq$), then you can actually factorize $N$, the attack basically works like the chosen ciphertext attack on Rabins encryption scheme: Choose $a$, calculate $a^2$, apply your algorithm to $a^2$ to find a square root. Since it only has $a^2$ as input, it will result in one of the four square roots, but not necessarily the one you started with. If the result is not $a$ or $-a$, then you can factorize $N$.


text:
# Is there a function such that the integral converges?

I would like to know if there are functions $f:\mathbb{R} \rightarrow \mathbb{R}$ (apart from the trivial one $f=0$) such that

$$\left|\int_{\mathbb{R}^n} f\left(\sum_{i=1}^n x_i\right)dx_1 \cdots dx_n\right|<\infty.$$

Also, whether such class of functions is too narrow.

My feeling is that it is not possible. Any suggestions or ideas?

Thanks! :)

• As a quick idea : $f(x) = \exp( - x^2 )$ should works – Tryss Nov 17 '15 at 20:54
• $f = 0$ comes to mind. – Robert Israel Nov 17 '15 at 20:54
• @Tryss No, that does not. – Robert Israel Nov 17 '15 at 20:57
• No it doesn't because it is the square of the sum and hence, there is a hyperplane where the exponent is 0 and hence the function becomes 1 and diverges... I can't think of a good example, not even $f$ with compact support seems to help. – Martingalo Nov 17 '15 at 20:59
• Martingalo : but the hyperplane is measure zero – Tryss Nov 17 '15 at 21:02

If you interpret the integral as a Lebesgue integral over $\mathbb R^n$, and $n \ge 2$, then you must have $f(x) = 0$ a.e. in order for the integral to converge.
Proof Suppose not. Then for some $\epsilon > 0$ there is a set $A\subset \mathbb R$ of nonzero measure where $|f| > \epsilon > 0$. If $\phi$ is the linear functional $(x_1, \ldots, x_n) \mapsto x_1 + \ldots + x_n$, then $|f(\sum_j x_j)| > \epsilon$ on $\phi^{-1}(A)$ which has infinite $n$-dimensional Lebesgue measure.
If it's an iterated improper (Riemann) integral, then any $f$ with $\int_{-\infty}^\infty f(x)\; dx = 0$ will do.


text:
# Hedwig: A Named Entity Linker

Marcus KlangPierre Nugues

Named entity linking is the task of identifying mentions of named things in text, such as {}Barack Obama{''} or {}New York{''}, and linking these mentions to unique identifiers. In this paper, we describe Hedwig, an end-to-end named entity linker, which uses a combination of word and character BILSTM models for mention detection, a Wikidata and Wikipedia-derived knowledge base with global information aggregated over nine language editions, and a PageRank algorithm for entity linking... (read more)

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