Actions and Strategies

Strategy

One's desire is either raising any suits value or getting any Three of a Kind (TOK). As this is not directly achievable, the goal is to maximize the score of a suit in hand or get two of a kind with one of the cards in hand. The intended path is highly dependent on both the cards in hand and the cards in the widow.

At the start of the game the players know very little of the strategies of the other players. Because of this is not possible to know what cards to take to block the others. This means the first exchange will be based on increasing the hand value and/or going towards TOK. With these moves a player would like to keep their options open as much as possible. When going for a TOK you will have to try to have the card which is not part of the TOK be in the same suit as one of the cards in the TOK. In general it may be necessary to change your strategy (and goal) as there is a significant chance that others will interrupt your intended course of action. Your choice of strategy will depend on several factors. The most obvious are your own cards. The widow’s cards might have cards initially which help with setting up your strategy. The strategies of other can also deter you from going for the same strategy as they are.

Once some actions have been taken the players are able to reason about the strategies of others. If a player knows the strategy of others they have to make a decision whether to try to block them or not. When there is no option to improve your own hand's value blocking a different player's from getting a full suit or TOK becomes a good strategy. Weakening one's own hand to block others is also an option but it has a lot of risk involved. When blocking a TOK this is a useful strategy as a TOK value is low if not all of the cards are there. When attempting to block someone you do need to evaluate if taking one card away is enough to block them. If there are already three high spades in the widow’s cards then it likely is a bad idea to block spades.

You can split the action up into two different decisions. The first one is “which card do I take” and the second one is “which card do I return to the widow”. The first decision is fairly simple, for increasing your own hand value you pick the card that works best with your strategy. It is not uncommon to have cards with similar value for a player. When this happens how much it influences their other goals. There might be a card that blocks as well or a card that leaves more options if someone blocks you later.

Deciding which card to return will be based on the general value of the card and the known strategies. First off you do not want to impede your own strategy. Then you see if any of the remaining cards fits into the other player's strategy. If this still does not result into a card to return you chose the lowest value one.

One is able to reason about possible strategies of a player based on the details of the card they exchange. This is a specific action. Some actions give more information than others. Most actions can be seen as both increasing their own hand value or trying to block others. A think to keep in mind is that if a action is done to block someone else it means that their strategy was known. Because all actions are public this means you would know their strategy as well, thus before the action takes place you can already know if blocking someone else could be a possible reason for that action.

(Rogchert Zijlstra , Hirad Emami Alagha)

Actions

Action (a1) Dropping a high card and picking a lower card in the same suit.

This action is a very rare one as it only fits with TOK strategies which are more rare than going for a suit. This also means that this move is one of the most telling moves you can make.

1. The player intends to raise his hand's highest suit value: Since the card is of the same suit but a lower value it is easy to see that that suits value drops. Therefore it makes no sense to make this action if you intend to raise your suit value.

2. The player wants to have three-of-a-kind: This intention is fitting for the action as it can not raise the value of the suit. If this action is taken in the first round then it's safe to assume that a TOK is the intend. As the intention is to get a TOK over suit value you can assume he already has one From this action you can reason about his remaining hand. First off, it might be the last card in the TOK. If this is the case that player will check the next round and other can't use this information to prevent this. However if this is not the case we know the following. The remaining card has the same suit as one of the cards in the TOK, otherwise it would be traded with the widow instead.

3. The player wants to block another player from raising his hand’s value: This is not possible. If ones intent is to prevent them from raising their hand you would not trade a higher card of that suit with the widow. Thus, it is not logically possible to block someone from raising his hand's value by dropping a high card.

4. The player wants to block another player from getting three-of-a-kind: This intend is possible. Especially strong if they were already going for that same suit in the first place. The player weakens his own hand to severely hamper the other. Though lowering your own hand might sound like a bad choice it can cause more damage to someone going for TOK as the value of just two parts is just the suit value.

Action (a3) Dropping a card and picking a higher card in the same suit.

This action is relatively rare in the beginning of the game as their are usually better options. Chances are high that if this action is taken then the player already has a strong hand.

1. The players intends to raise his hand's value: The player dropped a card to get a higher card of the same suit. Choosing to drop a card in the same suit for a higher one also tells us something about their other cards. As they intend to increase the value of that suit as high as possible the other cards in his hand are also of that suit and of higher value, otherwise they would be traded with the widow instead.

2: The player wants to have three-of-a-kind: Other player may also think this is a possibility. The player exchanges his card for a card with a new value. This value might be the one for which they intend to get a TOK. As with A1 one can assume that the card which is not part of the TOK is a card is the same suit as one of the cards in the TOK, otherwise it would be traded with the widow instead.

3. The player wants to block another player from raising his hand’s value: This can happen quite often. It might be a consequence of raising his own hand value in which case it was not the main reason for the action but still a beneficial side effect. This might also be the strategy when a player has 2 cards which belong to his strategy and one outlier, then it is a safe move to trade that outlier with a higher valued one in the widow unless someone is going for a TOK of that value.

4. The player wants to block another player from getting three-of-a-kind: As with blocking the suit value it can be a consequence of raising your own hands value. Otherwise it still is a strong move even if it does not benefit your own hand as messing up a TOK is often a very big loss for the one going for the TOK.

Checking: A player can check for a set of reasons. First off you can't have 31 as that already implies you win due to the game rules, the other players do not get a final turn. The best situation to be in is knowing that your cards are better than the opponent and what they could get within one round. In this case checking is a guaranteed win. Not all cards are played, and certainty on what the. In that case he has to ask himself "can I reach another goal". If this is not the case checking would be the best option. Lastly there is also the situation where you know the someone has checked with a higher score then you can possibly reach. This would mean an automatic loss no matter the outcome and checking is just a way to quicken the game ( This is also the reason why there are variants where winning does not matter, it only matter that you don't lose).

(Rogchert Zijlstra , Hirad Emami Alagha)

Action 2 (a2) Dropping a high card and picking a lower card in the different suite

One of the actions that an opponent can take is exchanging a card from his hand with a card that has a lower value and different suit. Once this action was taken by a player (for example player i); other players can consider four possibilities for the player i’s intention. The four possible intentions are listed below:

1. The player intends to raise his hand’s value: Since the dropped card has the different suit than the new card; it can be assumed that the player is raising his hand’s value that consists of the newly picked card’s suit. Thus, it can also be assumed that the player already owns a high value card with the same suit. Furthermore, even if the value of the card that player drops is not higher than the card that he picks; the same assumption can be made. This is due to two cards having different suits. Since the value of the hand is determined by the combined value of cards with same suit; the value of the card that player exchanges does not lead to different assumptions.

2. The player wants to have three-of-a-kind: Since the value of the card that the player exchanged was lower than the card he dropped, other players might consider the possibility that the player is aiming for a three-of-a-kind with the newly picked card’s value. This can lead to an assumption that the player has at least one card with same value as the newly picked card. In fact, even if the value of the card that player drops is not higher than the value of the card that he picks; other player would still consider that player i intends to complete his three of a kind hand. Thus, the value of the card does not affect the assumptions.

3. The player wants to block another player from raising his hand’s value: Other players can consider the possibility that the player wants to specifically block another player from raising his hand value of the picked card’s suit. However, other players would only make such assumption if and only if they know that another player (for example player j) has high cards of the suite that player i picked. Therefore, other players would not make such assumption in the first round when no one is aware of other players’ cards. In this case, the value of the card that player exchange does not affect the assumption. Thus, in both actions two and four, given the players’ knowledge; the same assumption can be made.

Assuming group G consists of all players except player i and π stands for having a high value card with the same suit as the card that player i picks

(M, s) |= E-BEL_G (INT (i, λ)) iff (M, s) |= Cp_jπ

4. The player wants to block another payer from getting three of a kind:

As another possible assumption, once the player i takes action two or four, other players can also believe that player one has the intention of blocking another player (player j) if and only if it is a common knowledge among players that player j holds at least one card with the same value of the card that player i picks.

Action 4 (a4) Dropping a low card and picking a higher card in the different suite

As it was mentioned in the previous section (a2), the results of action 4 (a4) is completely similar to a2. This is due to fact that dropping a lower valued card and picking a card with higher value, regardless of the suites; imply same intentions. Without having any prior knowledge of player's hand or previous intentions; there is no different in what actions of a2 and a4 would imply. Thus redundant explanation of a4 and the intentions it implies, is not included here.

Case 1: If player takes the action two in the first round

Description: In the first round all the players have limited knowledge about the cards that are used during this match. Each player only knows the cards that he has and three cards that are set for widow. Since none of the players is aware of other players’ cards; then it is not possible that he can have intentions of blocking someone else from raising his hand’s value (λ) or preventing others from having three of a kind (χ). Therefore; others can only assume that he either intends to raise his own hand’s value (φ) or get three of a kind (ψ).

In the rare case that one player (for example player j) starts the game with three of a kind (holding three cards with same value in different suits), then that one player would believe that player i intends to increase his hand’s value.

(Hirad Eami Alagha, Eric Dansen)